reserve x, r for Real;
reserve A for symmetrical Subset of COMPLEX;
reserve F,G for PartFunc of REAL, REAL;

theorem Th63:
  for x being Real holds absreal.(-x) = absreal.x
proof
  let x be Real;
  per cases;
  suppose
A1: x < 0;
    then absreal.(-x) = -x by Th61;
    hence thesis by A1,Th62;
  end;
  suppose
A2: 0 < x;
    then absreal.(-x) = -(-x) by Th62
      .=x;
    hence thesis by A2,Th61;
  end;
  suppose
    x = 0;
    hence thesis;
  end;
end;
