reserve a, b for Real,
  r for Real,
  rr for Real,
  i, j, n for Nat,
  M for non empty MetrSpace,
  p, q, s for Point of TOP-REAL 2,
  e for Point of Euclid 2,
  w for Point of Euclid n,
  z for Point of M,
  A, B for Subset of TOP-REAL n,
  P for Subset of TOP-REAL 2,
  D for non empty Subset of TOP-REAL 2;
reserve a, b for Real;

theorem
 for a,b being Real
  holds abs <*a,b*> = <*|.a.|,|.b.|*>
proof let a,b be Real;
  dom absreal = REAL by FUNCT_2:def 1;
  then rng <*a,b*> c= dom absreal;
  then
A1: dom abs <*a,b*> = dom <*a,b*> by RELAT_1:27
    .= {1,2} by FINSEQ_1:2,89;
A2: dom <*|.a.|,|.b.|*> = {1,2} by FINSEQ_1:2,89;
   reconsider a,b as Element of REAL by XREAL_0:def 1;
  for i being object st i in dom <*|.a.|,|.b.|*> holds (abs <*a,b*>).i = <*
  |.a.|,|.b.|*>.i
  proof
    let i be object;
A3: <*a,b*>.1 = a;
A4: <*a,b*>.2 = b;
A5: 2 in {1,2} by TARSKI:def 2;
A6: 1 in {1,2} by TARSKI:def 2;
    assume
A7: i in dom <*|.a.|,|.b.|*>;
    per cases by A2,A7,TARSKI:def 2;
    suppose
A8:   i = 1;
      hence (abs <*a,b*>).i = |.a.| by A1,A3,A6,Th11
        .= <*|.a.|,|.b.|*>.i by A8;
    end;
    suppose
A9:   i = 2;
      hence (abs <*a,b*>).i = |.b.| by A1,A4,A5,Th11
        .= <*|.a.|,|.b.|*>.i by A9;
    end;
  end;
  hence thesis by A1,FINSEQ_1:2,89,FUNCT_1:2;
end;
