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
  1_minus (f*g) = (1_minus f) ++ (1_minus g)
proof
A1: C = dom ((1_minus f) ++ (1_minus g)) by FUNCT_2:def 1;
A2: for c being Element of C st c in C holds (1_minus (f*g)).c = ((1_minus f
  ) ++ (1_minus g)).c
  proof
    let c;
    ((1_minus f) ++ (1_minus g)).c = (1_minus f).c + (1_minus g).c - ((
    1_minus f).c)*((1_minus g).c) by Def3
      .= 1 - f.c + (1_minus g).c - ((1_minus f).c)*((1_minus g).c) by
FUZZY_1:def 5
      .= 1 - f.c + (1- g.c) - ((1_minus f).c)*((1_minus g).c) by FUZZY_1:def 5
      .= 1 - f.c + (1- g.c) - (1 - f.c)*((1_minus g).c) by FUZZY_1:def 5
      .= 1 - f.c + (1- g.c) - (1 - f.c)*(1- g.c) by FUZZY_1:def 5
      .= g.c- g.c + 1 - f.c*g.c
      .= 1 - (f*g).c by Def2;
    hence thesis by FUZZY_1:def 5;
  end;
  C = dom 1_minus (f*g) by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
