reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th22:
  for A, B, C, D, E, F being set st A c= B & C c= D & E c= F holds
  product ((a,b,c) --> (A,C,E)) c= product ((a,b,c) --> (B,D,F))
  proof
    let A, B, C, D, E, F be set such that
A1: A c= B & C c= D & E c= F;
    set f = (a,b,c) --> (A,C,E), g = (a,b,c) --> (B,D,F);
A2: dom f = {a,b,c} & dom g = {a,b,c} by FUNCT_4:128;
    per cases;
    suppose a = c & a <> b;
      then f = (a,b) --> (E,C) & g = (a,b) --> (F,D) by FUNCT_4:132;
      hence product f c= product g by A1,TOPREAL6:21;
    end;
    suppose b = c & a <> b;
      then f = (a,b) --> (A,E) & g = (a,b) --> (B,F) by FUNCT_4:133;
      hence product f c= product g by A1,TOPREAL6:21;
    end;
    suppose a = b;
      then f = (a,c) --> (C,E) & g = (a,c) --> (D,F) by FUNCT_4:81;
      hence product f c= product g by A1,TOPREAL6:21;
    end;
    suppose
A3:   a <> b & a <> c & b <> c;
      for x being object st x in dom f holds f.x c= g.x
      proof
        let x be object;
        assume x in dom f;
        then
A4:     x = a or x = b or x = c by A2,ENUMSET1:def 1;
        a,b,c are_mutually_distinct by A3;
        then f.a = A & f.b = C & f.c = E & g.a = B & g.b = D & g.c = F
        by FUNCT_4:135,134;
        hence thesis by A1,A4;
      end;
      hence thesis by A2,CARD_3:27;
    end;
  end;
