theorem Th43:
  |.r qua Complex.| = |. r .|
proof
 reconsider rr=r as Real;
  per cases;
  suppose
A1: 0 <= r;
    then |. r .| = r by EXTREAL1:def 1;
    hence thesis by A1,ABSVALUE:def 1;
  end;
  suppose
A2: r < 0;
    then |. r .| =-(r qua ExtReal) by EXTREAL1:def 1;
    then |. rr .| =-rr;
    hence thesis by A2,ABSVALUE:def 1;
  end;
end;
