theorem Th37:
  <.F \/ H.) = <.F "/\" H.)
proof
  F c= F "/\" H & H c= F "/\" H by Th36;
  then F \/ H c= F "/\" H by XBOOLE_1:8;
  hence <.F \/ H.) c= <.F "/\" H.) by Th22;
  F"/\"H c= <.F \/ H.)
  proof
    let x be object;
    assume x in F"/\"H;
    then consider p,q such that
A1: x = p"/\"q and
A2: p in F and
A3: q in H;
    H c= F \/ H by XBOOLE_1:7;
    then
A4: q in F \/ H by A3;
A5: F \/ H c= <.F \/ H.) by Def4;
    F c= F \/ H by XBOOLE_1:7;
    then p in F \/ H by A2;
    hence thesis by A1,A4,A5,Th9;
  end;
  then <.F"/\"H.) c= <.<.F \/ H.).) by Th22;
  hence thesis by Th21;
end;
