theorem Th40:
  f is one-to-one & rng f = dom g & g*f = id dom f implies g = f"
proof
  assume that
A1: f is one-to-one and
A2: rng f = dom g & g*f = id dom f;
  f*f" = id rng f & rng(f") = dom f by A1,Th32,Th38;
  hence thesis by A2,Lm1;
end;
