reserve x,y,y1,y2 for set;
reserve C for non empty set;
reserve c for Element of C;
reserve f,h,g,h1 for Membership_Func of C;

theorem
  f c= g implies h\g c= h\f
proof
  assume
A1: f.c <= g.c;
  let c;
  f.c <= g.c by A1;
  then 1 - f.c >= 1 - g.c by XREAL_1:10;
  then (1_minus g).c <= 1- f.c by FUZZY_1:def 5;
  then (1_minus g).c <= (1_minus f).c by FUZZY_1:def 5;
  then min((1_minus g).c,h.c) <= min((1_minus f).c,h.c) by XXREAL_0:18;
  then (h\g).c <= min((1_minus f).c,h.c) by FUZZY_1:5;
  hence thesis by FUZZY_1:5;
end;
