theorem Th31:
  for P,A,B,C st C = B` & 0 < P.B & 0 < P.C holds P.A = P.|.B.A *
  P.B + P.|.C.A * P.C
proof
  let P,A,B,C;
  assume that
A1: C = B` and
A2: 0 < P.B and
A3: 0 < P.C;
  P.A = P.(A /\ B) + P.(A /\ C) by A1,Th14
    .= P.|.B.A * P.B + P.(A /\ C) by A2,Th29
    .= P.|.B.A * P.B + P.|.C.A * P.C by A3,Th29;
  hence thesis;
end;
