theorem Th34:
  (A + B) * C c= A * C + (B * C)
proof
  let x be object;
  assume x in (A + B) * C;
  then consider a,b such that
A1: x = a * b and
A2: a in A + B and
A3: b in C;
  consider g,h such that
A4: a = g + h & g in A and
A5: h in B by A2;
A6: h * b in B * C by A3,A5;
  x = (g * b) + (h * b) & g * b in A * C by A1,A3,A4,Th23;
  hence thesis by A6;
end;
