reserve x for set;
reserve a,b,c,d for ExtReal;

theorem
  c <= a & c <= b & (for d st d <= a & d <= b holds d <= c) implies c =
  min(a,b)
proof
  assume that
A1: c <= a & c <= b and
A2: for d st d <= a & d <= b holds d <= c;
  min(a,b) <= a & min(a,b) <= b by Th17;
  then
A3: min(a,b) <= c by A2;
  c <= min(a,b) by A1,Def8;
  hence thesis by A3,Th1;
end;
