reserve c,c1,c2,x,y,z,z1,z2 for set;
reserve C1,C2,C3 for non empty set;

theorem Th6:
  for f be RMembership_Func of C1,C2 holds 1_minus(converse f) =
  converse(1_minus f)
proof
  let f being RMembership_Func of C1,C2;
A1: [:C2,C1:] = dom converse(1_minus f ) by FUNCT_2:def 1;
A2: for c being Element of [:C2,C1:] st c in [:C2,C1:] holds (1_minus(
  converse f)).c = (converse(1_minus f)).c
  proof
    let c being Element of [:C2,C1:];
    assume c in [:C2,C1:];
    consider y,x being object such that
A3: y in C2 and
A4: x in C1 and
A5: c = [y,x] by ZFMISC_1:def 2;
    reconsider y,x as set by TARSKI:1;
A6: [x,y] in [:C1,C2:] by A3,A4,ZFMISC_1:87;
    (1_minus(converse f)).(y,x) = 1-(converse f).(y,x) by A5,FUZZY_1:def 5
      .= 1-f.(x,y) by A5,Def1
      .= (1_minus f).(x,y) by A6,FUZZY_1:def 5
      .= (converse(1_minus f)).(y,x) by A5,Def1;
    hence thesis by A5;
  end;
  dom(1_minus(converse f)) = [:C2,C1:] by FUNCT_2:def 1;
  hence thesis by A2,A1,PARTFUN1:5;
end;
