theorem
  (for x st x in B holds f.x [= g.x) implies FinJoin(B,f) [= FinJoin(B,g )
proof
  assume
A1: for x st x in B holds f.x [= g.x;
  now
    let x;
    assume
A2: x in B;
    then f.x [= g.x by A1;
    hence f.x [= FinJoin(B,g) by A2,Th29;
  end;
  hence thesis by Th54;
end;
