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

theorem
  max(max(min(a,b),min(b,c)),min(c,a)) = min(min(max(a,b),max(b,c)),max( c,a))
proof
  per cases;
  suppose
A1: a <= c;
    then
A2: max(a,b) <= max(b,c) by Th26;
    min(a,b) <= min(b,c) by A1,Th18;
    hence max(max(min(a,b),min(b,c)),min(c,a)) = max(min(b,c),min(c,a)) by
Def9
      .= max(min(b,c),a) by A1,Def8
      .= min(max(a,b),c) by A1,Th37
      .= min(max(a,b),max(c,a)) by A1,Def9
      .= min(min(max(a,b),max(b,c)),max(c,a)) by A2,Def8;
  end;
  suppose
A3: c <= a;
    then
A4: max(a,b) >= max(b,c) by Th26;
    min(a,b) >= min(b,c) by A3,Th18;
    hence max(max(min(a,b),min(b,c)),min(c,a)) = max(min(a,b),min(c,a)) by
Def9
      .= max(min(a,b),c) by A3,Def8
      .= min(max(c,b),a) by A3,Th37
      .= min(max(c,b),max(c,a)) by A3,Def9
      .= min(min(max(a,b),max(b,c)),max(c,a)) by A4,Def8;
  end;
end;
