theorem
  f is_integrable_on M & g is_integrable_on M implies f-g is_integrable_on M
proof
  assume that
A1: f is_integrable_on M and
A2: g is_integrable_on M;
A3: Im g is_integrable_on M by A2;
  Im f is_integrable_on M by A1;
  then Im(f)- Im(g) is_integrable_on M by A3,Th34;
  then
A4: Im(f-g) is_integrable_on M by Th6;
A5: Re g is_integrable_on M by A2;
  Re f is_integrable_on M by A1;
  then Re(f)- Re(g) is_integrable_on M by A5,Th34;
  then Re(f-g) is_integrable_on M by Th6;
  hence thesis by A4;
end;
