theorem Th45:
  a in C* & b in C* implies a ^ b in C*
proof
  assume that
A1: a in C* and
A2: b in C*;
  consider k such that
A3: a in C |^ k by A1,Th41;
  consider l such that
A4: b in C |^ l by A2,Th41;
  a ^ b in C |^ (k + l) by A3,A4,Th40;
  hence thesis by Th41;
end;
