% Mizar ND problem: t7_rfunct_3,rfunct_3,157,18 
fof(dh_c1_8__rfunct_3,definition,
    ( ( v1_xreal_0(c1_8__rfunct_3)
     => ! [A] : 
          ( v1_xreal_0(A)
         => ! [B] : 
              ( v1_xreal_0(B)
             => ! [C] : 
                  ( v1_xreal_0(C)
                 => ( ( r1_xreal_0(c1_8__rfunct_3,B)
                      & r1_xreal_0(A,C) )
                   => r1_xreal_0(k2_square_1(c1_8__rfunct_3,A),k2_square_1(B,C)) ) ) ) ) )
   => ! [D] : 
        ( v1_xreal_0(D)
       => ! [E] : 
            ( v1_xreal_0(E)
           => ! [F] : 
                ( v1_xreal_0(F)
               => ! [G] : 
                    ( v1_xreal_0(G)
                   => ( ( r1_xreal_0(D,F)
                        & r1_xreal_0(E,G) )
                     => r1_xreal_0(k2_square_1(D,E),k2_square_1(F,G)) ) ) ) ) ) ),
    introduced(definition,[new_symbol(c1_8__rfunct_3),file(rfunct_3,c1_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c1_8__rfunct_3)]).

fof(dh_c2_8__rfunct_3,definition,
    ( ( v1_xreal_0(c2_8__rfunct_3)
     => ! [A] : 
          ( v1_xreal_0(A)
         => ! [B] : 
              ( v1_xreal_0(B)
             => ( ( r1_xreal_0(c1_8__rfunct_3,A)
                  & r1_xreal_0(c2_8__rfunct_3,B) )
               => r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(A,B)) ) ) ) )
   => ! [C] : 
        ( v1_xreal_0(C)
       => ! [D] : 
            ( v1_xreal_0(D)
           => ! [E] : 
                ( v1_xreal_0(E)
               => ( ( r1_xreal_0(c1_8__rfunct_3,D)
                    & r1_xreal_0(C,E) )
                 => r1_xreal_0(k2_square_1(c1_8__rfunct_3,C),k2_square_1(D,E)) ) ) ) ) ),
    introduced(definition,[new_symbol(c2_8__rfunct_3),file(rfunct_3,c2_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c2_8__rfunct_3)]).

fof(dh_c3_8__rfunct_3,definition,
    ( ( v1_xreal_0(c3_8__rfunct_3)
     => ! [A] : 
          ( v1_xreal_0(A)
         => ( ( r1_xreal_0(c1_8__rfunct_3,c3_8__rfunct_3)
              & r1_xreal_0(c2_8__rfunct_3,A) )
           => r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(c3_8__rfunct_3,A)) ) ) )
   => ! [B] : 
        ( v1_xreal_0(B)
       => ! [C] : 
            ( v1_xreal_0(C)
           => ( ( r1_xreal_0(c1_8__rfunct_3,B)
                & r1_xreal_0(c2_8__rfunct_3,C) )
             => r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(B,C)) ) ) ) ),
    introduced(definition,[new_symbol(c3_8__rfunct_3),file(rfunct_3,c3_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c3_8__rfunct_3)]).

fof(dh_c4_8__rfunct_3,definition,
    ( ( v1_xreal_0(c4_8__rfunct_3)
     => ( ( r1_xreal_0(c1_8__rfunct_3,c3_8__rfunct_3)
          & r1_xreal_0(c2_8__rfunct_3,c4_8__rfunct_3) )
       => r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3)) ) )
   => ! [A] : 
        ( v1_xreal_0(A)
       => ( ( r1_xreal_0(c1_8__rfunct_3,c3_8__rfunct_3)
            & r1_xreal_0(c2_8__rfunct_3,A) )
         => r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(c3_8__rfunct_3,A)) ) ) ),
    introduced(definition,[new_symbol(c4_8__rfunct_3),file(rfunct_3,c4_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c4_8__rfunct_3)]).

fof(e1_8__rfunct_3,assumption,
    ( r1_xreal_0(c1_8__rfunct_3,c3_8__rfunct_3)
    & r1_xreal_0(c2_8__rfunct_3,c4_8__rfunct_3) ),
    introduced(assumption,[file(rfunct_3,e1_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,e1_8__rfunct_3)]).

fof(rc1_xreal_0,theorem,(
    ? [A] : 
      ( v1_xcmplx_0(A)
      & v1_xreal_0(A) ) ),
    file(xreal_0,rc1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,rc1_xreal_0)]).

fof(commutativity_k2_square_1,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => k2_square_1(A,B) = k2_square_1(B,A) ) ),
    file(square_1,k2_square_1),
    [interesting(0.9),axiom,file(square_1,k2_square_1)]).

fof(idempotence_k2_square_1,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => k2_square_1(A,A) = A ) ),
    file(square_1,k2_square_1),
    [interesting(0.9),axiom,file(square_1,k2_square_1)]).

fof(reflexivity_r1_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => r1_xreal_0(A,A) ) ),
    file(xreal_0,r1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]).

fof(connectedness_r1_xreal_0,theorem,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => ( r1_xreal_0(A,B)
        | r1_xreal_0(B,A) ) ) ),
    file(xreal_0,r1_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,r1_xreal_0)]).

fof(dt_k2_square_1,axiom,(
    ! [A,B] : 
      ( ( v1_xreal_0(A)
        & v1_xreal_0(B) )
     => v1_xreal_0(k2_square_1(A,B)) ) ),
    file(square_1,k2_square_1),
    [interesting(0.9),axiom,file(square_1,k2_square_1)]).

fof(dt_c1_8__rfunct_3,assumption,(
    v1_xreal_0(c1_8__rfunct_3) ),
    introduced(assumption,[file(rfunct_3,c1_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c1_8__rfunct_3)]).

fof(dt_c2_8__rfunct_3,assumption,(
    v1_xreal_0(c2_8__rfunct_3) ),
    introduced(assumption,[file(rfunct_3,c2_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c2_8__rfunct_3)]).

fof(dt_c3_8__rfunct_3,assumption,(
    v1_xreal_0(c3_8__rfunct_3) ),
    introduced(assumption,[file(rfunct_3,c3_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c3_8__rfunct_3)]).

fof(dt_c4_8__rfunct_3,assumption,(
    v1_xreal_0(c4_8__rfunct_3) ),
    introduced(assumption,[file(rfunct_3,c4_8__rfunct_3)]),
    [interesting(0.8),axiom,file(rfunct_3,c4_8__rfunct_3)]).

fof(cc2_xreal_0,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => v1_xcmplx_0(A) ) ),
    file(xreal_0,cc2_xreal_0),
    [interesting(0.9),axiom,file(xreal_0,cc2_xreal_0)]).

fof(t46_square_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => r1_xreal_0(A,k2_square_1(A,B)) ) ) ),
    file(square_1,t46_square_1),
    [interesting(0.9),axiom,file(square_1,t46_square_1)]).

fof(e2_8__rfunct_3,plain,
    ( r1_xreal_0(c3_8__rfunct_3,k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3))
    & r1_xreal_0(c4_8__rfunct_3,k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c3_8__rfunct_3,dt_c4_8__rfunct_3])],[rc1_xreal_0,commutativity_k2_square_1,idempotence_k2_square_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k2_square_1,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3,cc2_xreal_0,t46_square_1]),
    [interesting(0.8),file(rfunct_3,e2_8__rfunct_3),[file(rfunct_3,e2_8__rfunct_3)]]).

fof(t2_xreal_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(B,C) )
               => r1_xreal_0(A,C) ) ) ) ) ),
    file(xreal_1,t2_xreal_1),
    [interesting(0.9),axiom,file(xreal_1,t2_xreal_1)]).

fof(e3_8__rfunct_3,plain,
    ( r1_xreal_0(c1_8__rfunct_3,k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3))
    & r1_xreal_0(c2_8__rfunct_3,k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3,e1_8__rfunct_3])],[rc1_xreal_0,commutativity_k2_square_1,idempotence_k2_square_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k2_square_1,dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3,cc2_xreal_0,e2_8__rfunct_3,e1_8__rfunct_3,t2_xreal_1]),
    [interesting(0.8),file(rfunct_3,e3_8__rfunct_3),[file(rfunct_3,e3_8__rfunct_3)]]).

fof(t50_square_1,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(C,B) )
              <=> r1_xreal_0(k2_square_1(A,C),B) ) ) ) ) ),
    file(square_1,t50_square_1),
    [interesting(0.9),axiom,file(square_1,t50_square_1)]).

fof(e4_8__rfunct_3,plain,(
    r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3)) ),
    inference(mizar_by,[status(thm),assumptions([dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3,e1_8__rfunct_3])],[rc1_xreal_0,commutativity_k2_square_1,idempotence_k2_square_1,reflexivity_r1_xreal_0,connectedness_r1_xreal_0,dt_k2_square_1,dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3,cc2_xreal_0,e3_8__rfunct_3,t50_square_1]),
    [interesting(0.8),file(rfunct_3,e4_8__rfunct_3),[file(rfunct_3,e4_8__rfunct_3)]]).

fof(i3_8__rfunct_3,theorem,(
    $true ),
    introduced(tautology,[file(rfunct_3,i3_8__rfunct_3)]),
    [interesting(0.8),trivial,file(rfunct_3,i3_8__rfunct_3)]).

fof(i2_8__rfunct_3,plain,(
    r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3)) ),
    inference(conclusion,[status(thm),assumptions([dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3,e1_8__rfunct_3])],[e4_8__rfunct_3,i3_8__rfunct_3]),
    [interesting(0.8),file(rfunct_3,i2_8__rfunct_3),[file(rfunct_3,i2_8__rfunct_3)]]).

fof(i1_8__rfunct_3,plain,
    ( ( r1_xreal_0(c1_8__rfunct_3,c3_8__rfunct_3)
      & r1_xreal_0(c2_8__rfunct_3,c4_8__rfunct_3) )
   => r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3)) ),
    inference(discharge_asm,[status(thm),assumptions([dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3]),discharge_asm(discharge,[e1_8__rfunct_3])],[e1_8__rfunct_3,i2_8__rfunct_3]),
    [interesting(0.8),file(rfunct_3,i1_8__rfunct_3),[file(rfunct_3,i1_8__rfunct_3)]]).

fof(i1_8_tmp__rfunct_3,plain,
    ( ( v1_xreal_0(c1_8__rfunct_3)
      & v1_xreal_0(c2_8__rfunct_3)
      & v1_xreal_0(c3_8__rfunct_3)
      & v1_xreal_0(c4_8__rfunct_3) )
   => ( ( r1_xreal_0(c1_8__rfunct_3,c3_8__rfunct_3)
        & r1_xreal_0(c2_8__rfunct_3,c4_8__rfunct_3) )
     => r1_xreal_0(k2_square_1(c1_8__rfunct_3,c2_8__rfunct_3),k2_square_1(c3_8__rfunct_3,c4_8__rfunct_3)) ) ),
    inference(discharge_asm,[status(thm),assumptions([]),discharge_asm(discharge,[dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3])],[dt_c1_8__rfunct_3,dt_c2_8__rfunct_3,dt_c3_8__rfunct_3,dt_c4_8__rfunct_3,i1_8__rfunct_3]),
    [interesting(1),t7_rfunct_3]).

fof(t7_rfunct_3,theorem,(
    ! [A] : 
      ( v1_xreal_0(A)
     => ! [B] : 
          ( v1_xreal_0(B)
         => ! [C] : 
              ( v1_xreal_0(C)
             => ! [D] : 
                  ( v1_xreal_0(D)
                 => ( ( r1_xreal_0(A,C)
                      & r1_xreal_0(B,D) )
                   => r1_xreal_0(k2_square_1(A,B),k2_square_1(C,D)) ) ) ) ) ) ),
    inference(let,[status(thm),assumptions([])],[i1_8_tmp__rfunct_3,dh_c1_8__rfunct_3,dh_c2_8__rfunct_3,dh_c3_8__rfunct_3,dh_c4_8__rfunct_3]),
    [interesting(1),file(rfunct_3,t7_rfunct_3),[file(rfunct_3,t7_rfunct_3)]]).