theorem Th6:
  for X being real-membered set holds (for r st r in X holds r<=g1
) & (for s st 0<s ex r st (r in X & g1-s<r)) & (for r st r in X holds r<=g2) &
  (for s st 0<s ex r st (r in X & g2-s<r)) implies g1=g2
proof
  let X be real-membered set;
  assume that
A1: for r st r in X holds r<=g1 and
A2: for s st 0<s ex r st r in X & g1-s<r and
A3: for r st r in X holds r<=g2 and
A4: for s st 0<s ex r st r in X & g2-s<r;
A5: now
    assume g2<g1;
    then ex r1 st r1 in X & g1-(g1-g2)<r1 by A2,XREAL_1:50;
    hence contradiction by A3;
  end;
  now
    assume g1<g2;
    then ex r1 st r1 in X & g2-(g2-g1)<r1 by A4,XREAL_1:50;
    hence contradiction by A1;
  end;
  hence thesis by A5,XXREAL_0:1;
end;
