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 Th36:
  f ++ g = 1_minus((1_minus f)*(1_minus g))
proof
A1: C = dom 1_minus((1_minus f)*(1_minus g)) by FUNCT_2:def 1;
A2: for c being Element of C st c in C holds (f ++ g).c = (1_minus((1_minus
  f)*(1_minus g))).c
  proof
    let c;
    (1_minus((1_minus f)*(1_minus g))).c = 1 - ((1_minus f)*(1_minus g)).c
    by FUZZY_1:def 5
      .= 1 - ((1_minus f).c)*((1_minus g).c) by Def2
      .= 1 - (1 - f.c)*((1_minus g).c) by FUZZY_1:def 5
      .= 1 - (1 - f.c)*(1 - g.c) by FUZZY_1:def 5
      .= f.c + g.c - f.c*g.c;
    hence thesis by Def3;
  end;
  C = dom (f ++ g) by FUNCT_2:def 1;
  hence thesis by A1,A2,PARTFUN1:5;
end;
