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

theorem
  min(min(a,b),c) = min(a,min(b,c))
proof
  per cases by Th2;
  suppose
    a <= b & a <= c;
    then min(a,b) = a & min(a,c) = a by Def8;
    hence thesis by Def8;
  end;
  suppose
    b <= a & b <= c;
    then min(a,b) = b & min(b,c) = b by Def8;
    hence thesis;
  end;
  suppose
    c <= b & c <= a;
    then min(b,c) = c & min(a,c) = c by Def8;
    hence thesis by Def8;
  end;
end;
