theorem Th116:
  rng f c= dom (g|S) implies (g|S)/*f = g/*f
proof
  assume
A1: rng f c= dom (g|S);
  let x be Element of X;
A2: dom (g|S) c= dom g by RELAT_1:60;
A3: f.x in rng f by Th4;
  thus ((g|S)/*f).x = (g|S).(f.x) by A1,Th107
    .= g.(f.x) by A1,A3,FUNCT_1:47
    .= (g/*f).x by A1,A2,Th107,XBOOLE_1:1;
end;
