 reserve a,b,c,x for Real;
 reserve C for non empty set;

theorem AlphaCut1:
  for F being FuzzySet of C,
      alpha being Real holds
    alpha-cut F = F " ([. alpha, 1 .])
  proof
    let F be FuzzySet of C,
        alpha be Real;
    thus alpha-cut F c= F " ([. alpha, 1 .])
    proof
      let x be object;
      assume x in alpha-cut F; then
      consider y being Element of C such that
A1:   x = y & F.y >= alpha;
      x in C by A1; then
A2:   x in dom F by FUNCT_2:def 1; then
      F.y in [. 0, 1 .] by A1,PARTFUN1:4; then
      0 <= F.y & F.y <= 1 by XXREAL_1:1; then
      F.y in [. alpha, 1 .] by A1;
      hence thesis by A1,FUNCT_1:def 7,A2;
    end;
    let x be object;
    assume
c2: x in F"([. alpha, 1 .]); then
    x in dom F & F.x in [. alpha, 1 .] by FUNCT_1:def 7; then
    alpha <= F.x & F.x <= 1 by XXREAL_1:1;
    hence thesis by c2;
  end;
