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 Th16:
  F is commutative associative & F is having_a_unity & H is
  commutative associative & H is having_a_unity & h.the_unity_wrt F =
the_unity_wrt H & (for d1,d2 holds h.(F.(d1,d2)) = H.(h.d1,h.d2)) implies h.(F
  $$(B,f)) = H $$(B,h*f)
proof
  assume that
A1: F is commutative associative & F is having_a_unity and
A2: H is commutative associative & H is having_a_unity and
A3: h.the_unity_wrt F = the_unity_wrt H and
A4: for d1,d2 holds h.(F.(d1,d2)) = H.(h.d1,h.d2);
  defpred X[Element of Fin C] means h.(F$$($1,f)) = H $$($1,h*f);
A5: 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 B,c such that
A6: h.(F$$(B,f)) = H $$(B,h*f) and
A7: not c in B;
    thus h.(F$$(B \/ {.c.},f)) = h.(F.(F$$(B,f),f.c)) by A1,A7,Th2
      .= H.(H $$(B,h*f),h.(f.c)) by A4,A6
      .= H.(H $$(B,h*f),(h*f).c) by FUNCT_2:15
      .= H $$(B \/ {.c.},h*f) by A2,A7,Th2;
  end;
  h.(F$$({}.C,f)) = the_unity_wrt H by A1,A3,SETWISEO:31
    .= H $$({}.C,h*f) by A2,SETWISEO:31;
  then
A8: X[{}.C];
  for B holds X[B] from SETWISEO:sch 2(A8,A5);
  hence thesis;
end;
