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 = (<->f) <#> g
proof
  set f1 = <->f;
A1: dom f1 = dom f & dom(f<#>-g) = dom f /\ dom -g by Def33,Def43;
  dom(f1<#>g) = dom f1 /\ dom g by Def43;
  hence
A2: dom(f<#>-g) = dom(f1<#>g) by A1,VALUED_1:8;
  let x be object;
  assume
A3: x in dom(f<#>-g);
  then
A4: x in dom f1 by A1,XBOOLE_0:def 4;
  thus (f<#>-g).x = f.x (#) (-g).x by A3,Def43
    .= f.x(#)-g.x by VALUED_1:8
    .= (-f.x)(#)g.x by Th22
    .= f1.x(#)g.x by A4,Def33
    .= (f1<#>g).x by A2,A3,Def43;
end;
