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 Th10:
 for a,b being Real holds sqr <*a,b*> = <*a^2,b^2*>
proof let a,b be Real;
  dom sqrreal = REAL by FUNCT_2:def 1;
  then
A1: rng <*a,b*> c= dom sqrreal;
A2: dom <*a^2,b^2*> = {1,2} by FINSEQ_1:2,89;
A3: for i being object st i in dom <*a^2,b^2*> holds (sqr <*a,b*>).i = <*a^2,b
  ^2*>.i
  proof
    let i be object;
A4: <*a,b*>.1 = a;
A5: <*a,b*>.2 = b;
    assume
A6: i in dom <*a^2,b^2*>;
    per cases by A2,A6,TARSKI:def 2;
    suppose
A7:   i = 1;
      hence (sqr <*a,b*>).i = a^2 by A4,VALUED_1:11
        .= <*a^2,b^2*>.i by A7;
    end;
    suppose
A8:   i = 2;
      hence (sqr <*a,b*>).i = b^2 by A5,VALUED_1:11
        .= <*a^2,b^2*>.i by A8;
    end;
  end;
  dom sqr <*a,b*> = dom (sqrreal*<*a,b*>) by RVSUM_1:def 8
    .= dom <*a,b*> by A1,RELAT_1:27
    .= {1,2} by FINSEQ_1:2,89;
  hence thesis by A3,FINSEQ_1:2,89,FUNCT_1:2;
end;
