reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;

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;
