reserve x,y,z,x1,x2,y1,y2,z1,z2 for object,
  i,j,k,l,n,m for Nat,
  D for non empty set,
  K for Ring;

theorem Th26:
  for I, J being non empty set for F,G being BinOp of D for f
being Function of I,D for g being Function of J,D for X being Element of Fin I
  for Y being Element of Fin J st F is having_a_unity & F is commutative & F is
associative & F is having_an_inverseOp & G is_distributive_wrt F holds F$$([:X,
  Y:],G*(f,g))=F$$(X,G[:](f,F$$(Y,g)))
proof
  let I, J be non empty set;
  let F,G be BinOp of D;
  let f be Function of I,D;
  let g be Function of J,D;
  let X be Element of Fin I;
  let Y be Element of Fin J;
  assume that
A1: F is having_a_unity & F is commutative & F is associative and
A2: F is having_an_inverseOp & G is_distributive_wrt F;
  defpred P[Element of Fin I] means F$$([:$1,Y:],G*(f,g))=F$$($1,G[:](f,F$$(Y,
  g)));
A3: for X1 being Element of Fin I,x being Element of I st P[X1] holds P[X1
  \/ {.x.}]
  proof
    let X1 be Element of Fin I,x be Element of I;
    reconsider s={.x.} as Element of Fin I;
    assume
A4: F$$([:X1,Y:],G*(f,g))=F$$(X1,G[:](f,F$$(Y,g)));
    now
      per cases;
      case
        x in X1;
        then X1 \/ {x} = X1 by ZFMISC_1:40;
        hence thesis by A4;
      end;
      case
        not x in X1;
        then
A5:     X1 misses {x} by ZFMISC_1:50;
        then
A6:     [:X1,Y:] misses [:s,Y:] by ZFMISC_1:104;
        thus F$$([:X1 \/ {.x.},Y:],G*(f,g)) =F$$([:X1,Y:] \/ [:s,Y:],G*(f,g))
        by ZFMISC_1:97
          .=F.(F $$([:X1,Y:],G*(f,g)),F $$([:s,Y:],G*(f,g))) by A1,A6,
SETWOP_2:4
          .=F.(F$$(X1,G[:](f,F$$(Y,g))),F$$(s,G[:](f,F$$(Y,g)))) by A1,A2,A4
,Th25
          .=F$$(X1 \/ {.x.},G[:](f,F$$(Y,g))) by A1,A5,SETWOP_2:4;
      end;
    end;
    hence thesis;
  end;
A7: P[{}.I]
  proof
    reconsider T={}.[:I,J:] as Element of Fin [:I,J:];
    T=[:{}.I,Y:] by ZFMISC_1:90;
    then F $$([:{}.I,Y:],G*(f,g))= the_unity_wrt F by A1,SETWISEO:31;
    hence thesis by A1,SETWISEO:31;
  end;
  for X1 being Element of Fin I holds P[X1] from SETWISEO:sch 4(A7,A3 );
  hence thesis;
end;
