reserve x,y for set;
reserve C,C9,D,E for non empty set;
reserve c,c9,c1,c2,c3 for Element of C;
reserve B,B9,B1,B2 for Element of Fin C;
reserve A for Element of Fin C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve f,f9 for Function of C,D;
reserve g for Function of C9,D;
reserve H for BinOp of E;
reserve h for Function of D,E;
reserve i,j for Nat;
reserve s for Function;
reserve p,q for FinSequence of D;
reserve T1,T2 for Element of i-tuples_on D;

theorem Th12:
  F is commutative associative & F is having_a_unity & e =
the_unity_wrt F & G is_distributive_wrt F & G.(d,e) = e implies G.(d,F$$(B,f))
  = F $$(B,G[;](d,f))
proof
  assume that
A1: F is commutative associative & F is having_a_unity and
A2: e = the_unity_wrt F and
A3: G is_distributive_wrt F;
  defpred X[Element of Fin C] means G.(d,F$$($1,f)) = F $$($1,G[;](d,f));
A4: for B9 being Element of Fin C, b being Element of C holds X[B9] & not b
  in B9 implies X[B9 \/ {.b.}]
  proof
    let B9,c such that
A5: G.(d,F$$(B9,f)) = F $$(B9,G[;](d,f)) and
A6: not c in B9;
    thus G.(d,F$$(B9 \/ {.c.},f)) = G.(d,F.(F$$(B9,f),f.c)) by A1,A6,Th2
      .= F.(G.(d,F$$(B9,f)),G.(d,f.c)) by A3,BINOP_1:11
      .= F.(F $$(B9,G[;](d,f)),(G[;](d,f)).c) by A5,FUNCOP_1:53
      .= F $$(B9 \/ {.c.},G[;](d,f)) by A1,A6,Th2;
  end;
  assume G.(d,e) = e;
  then G.(d,F$$({}.C,f)) = e by A1,A2,SETWISEO:31
    .= F $$({}.C,G[;](d,f)) by A1,A2,SETWISEO:31;
  then
A7: X[{}.C];
  for B holds X[B] from SETWISEO:sch 2(A7,A4);
  hence thesis;
end;
