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 Th28:
  f*f c= f & f c= f ++ f
proof
  thus f*f c= f
  proof
    let c;
A1: 0 <= f.c by Th1;
    f.c <= 1 by Th1;
    then (f.c)*(f.c) <= 1*(f.c) by A1,XREAL_1:64;
    hence thesis by Def2;
  end;
  let c;
A2: 0 <= f.c by Th1;
  0 <= (1_minus f).c by Th1;
  then 0*(f.c) <= (f.c)*((1_minus f).c) by A2,XREAL_1:64;
  then 0 <= (f.c)*(1 - f.c) by FUZZY_1:def 5;
  then 0 + f.c <= f.c - (f.c)*(f.c) + f.c by XREAL_1:6;
  then f.c <= f.c + f.c - (f.c)*(f.c);
  hence thesis by Def3;
end;
