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

theorem Th18:
  a <= b & c <= d implies min(a,c) <= min(b,d)
proof
  assume that
A1: a <= b and
A2: c <= d;
  min(a,c) <= c by Th17;
  then
A3: min(a,c) <= d by A2,Th2;
  min(a,c) <= a by Th17;
  then min(a,c) <= b by A1,Th2;
  hence thesis by A3,Def8;
end;
