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) by Def2
      .= (1 - f.c)*((1_minus g).c) by FUZZY_1:def 5
      .= (1 - f.c)*(1- g.c) by FUZZY_1:def 5
      .= 1 - (f.c+ g.c - f.c*g.c)
      .= 1 - (f ++ g).c by Def3;
    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;
