reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;

theorem Th26:
  for A, X, Y being non empty set for F being BinOp of A st F is
  idempotent & F is commutative & F is associative for B being Element of Fin X
st B <> {} for f being Function of X,Y holds for g being Function of Y,A holds
  F$$(f.:B,g) = F$$(B,g*f)
proof
  let A, X, Y be non empty set, F be BinOp of A such that
A1: F is idempotent and
A2: F is commutative & F is associative;
  let B be Element of Fin X such that
A3: B <> {};
  let f be Function of X,Y;
  let g be Function of Y,A;
  defpred X[Element of Fin X] means F$$(f.:$1,g) = F$$($1,g*f);
A4: dom f = X by FUNCT_2:def 1;
A5: for B1,B2 being non empty Element of Fin X holds X[B1] & X[B2] implies X
  [B1 \/ B2]
  proof
    let B1,B2 be non empty Element of Fin X;
    assume
A6: F$$(f.:B1,g) = F$$(B1,g*f) & F$$(f.:B2,g) = F$$(B2,g*f);
A7: B1 c= X by FINSUB_1:def 5;
A8: B2 c= X by FINSUB_1:def 5;
    thus F$$(f.:(B1 \/ B2),g) = F$$(f.:B1 \/ f.:B2, g) by RELAT_1:120
      .= F.(F$$(f.:B1,g), F$$(f.:B2,g)) by A1,A2,A7,A8,Th18,A4
      .= F$$(B1 \/ B2,g*f) by A1,A2,A6,Th18;
  end;
A9: for x being Element of X holds X[{.x.}]
  proof
    let x be Element of X;
    f.:{.x.} = Im(f,x);
    hence F$$(f.:{.x.},g) = F$$({.f.x.},g) by A4,FUNCT_1:59
      .= g.(f.x) by A2,Th14
      .= (g*f).x by FUNCT_2:15
      .= F$$({.x.},g*f) by A2,Th14;
  end;
  for B being non empty Element of Fin X holds X[B] from FinSubInd2(A9,
  A5);
  hence thesis by A3;
end;
