theorem Th23:
  f is_integrable_on M implies f|A is_integrable_on M
proof
  assume
A1: f is_integrable_on M;
  then Im f is_integrable_on M;
  then (Im f)|A is_integrable_on M by MESFUNC6:91;
  then
A2: Im(f|A) is_integrable_on M by Th7;
  Re f is_integrable_on M by A1;
  then (Re f)|A is_integrable_on M by MESFUNC6:91;
  then Re(f|A) is_integrable_on M by Th7;
  hence thesis by A2;
end;
