reserve x,y,w,z for ExtReal,
  a for Real;

theorem Th1:
  x = a implies |.x.| = |.a qua Complex.|
proof
  assume
A1: x = a;
  reconsider x as R_eal by XXREAL_0:def 1;
  per cases;
  suppose
 0 <= x;
    then |.x.| = a by A1,Def1;
    hence thesis by ABSVALUE:def 1;
  end;
  suppose
A2: not 0 <= x;
    then |.x.| = -x by Def1
      .= -a by A1,SUPINF_2:2;
    hence thesis by A1,A2,ABSVALUE:def 1;
  end;
end;
