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
  f <#> g </> h = f <#> (g/"h)
proof
  set f1 = f<#>g;
A1: dom(g/"h) = dom g /\ dom h by VALUED_1:16;
A2: dom(f1</>h) = dom f1 /\ dom h by Th71;
  dom f1 = dom f /\ dom g & dom(f<#>(g/"h)) = dom f /\ dom(g/"h) by Def43;
  hence
A3: dom(f1</>h) = dom(f<#>(g/"h)) by A2,A1,XBOOLE_1:16;
  let x be object;
  assume
A4: x in dom(f1</>h);
  then
A5: x in dom f1 by A2,XBOOLE_0:def 4;
  thus (f1</>h).x = f1.x (/) h.x by A4,Th72
    .= f.x (#) g.x (/) h.x by A5,Def43
    .= f.x (#) (g.x / h.x) by Th16
    .= f.x (#) ((g/"h).x) by VALUED_1:17
    .= (f<#>(g/"h)).x by A3,A4,Def43;
end;
