theorem Th11:
  F \/ G on P iff F on P & G on P
proof
  thus F \/ G on P implies F on P & G on P by Th7,XBOOLE_1:7;
  assume
A1: F on P & G on P;
  let C be POINT of S;
  assume C in F \/ G;
  then C in F or C in G by XBOOLE_0:def 3;
  hence thesis by A1;
end;
