reserve x, y for object, X, X1, X2 for set;
reserve Y, Y1, Y2 for complex-functions-membered set,
  c, c1, c2 for Complex,
  f for PartFunc of X,Y,
  f1 for PartFunc of X1,Y1,
  f2 for PartFunc of X2, Y2,
  g, h, k for complex-valued Function;

theorem Th89:
  f <##> (f1<-->f2) = (f<##>f1) <--> (f<##>f2)
proof
  set f3 = f<##>f1, f4 = f<##>f2, f5 = f1<-->f2;
A1: dom(f<##>f5) = dom f /\ dom f5 by Def47;
A2: dom f5 = dom f1 /\ dom f2 by Def46;
A3: dom(f3<-->f4) = dom f3 /\ dom f4 by Def46;
  dom f3 = dom f /\ dom f1 & dom f4 = dom f /\ dom f2 by Def47;
  hence
A4: dom(f<##>f5) = dom(f3<-->f4) by A1,A3,A2,Lm1;
  let x be object;
  assume
A5: x in dom(f<##>f5);
  then
A6: x in dom f3 by A3,A4,XBOOLE_0:def 4;
A7: x in dom f5 by A1,A5,XBOOLE_0:def 4;
A8: x in dom f4 by A3,A4,A5,XBOOLE_0:def 4;
  thus (f<##>f5).x = f.x (#) f5.x by A5,Def47
    .= f.x (#) (f1.x - f2.x) by A7,Def46
    .= f.x (#) f1.x - f.x (#) f2.x by RFUNCT_1:15
    .= f3.x - f.x (#) f2.x by A6,Def47
    .= f3.x - f4.x by A8,Def47
    .= (f3<-->f4).x by A4,A5,Def46;
end;
