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 Th39:
  1_minus(EMF(C)) = UMF(C) & 1_minus(UMF(C)) = EMF(C)
proof
A1: for x being Element of C st x in C holds (1_minus(EMF(C))).x = (UMF(C)). x
  proof
    let x be Element of C;
    (1_minus(EMF(C))).x = 1 - (EMF(C)).x by Def5
      .= 1 - 0 by FUNCT_3:def 3
      .= 1;
    hence thesis by FUNCT_3:def 3;
  end;
  C = dom 1_minus(EMF(C)) & C = dom UMF(C) by FUNCT_2:def 1;
  hence 1_minus(EMF(C)) = UMF(C) by A1,PARTFUN1:5;
A2: for x being Element of C st x in C holds (1_minus(UMF(C))).x = (EMF(C)). x
  proof
    let x be Element of C;
    (1_minus(UMF(C))).x = 1 - (UMF(C)).x by Def5
      .= 1 - 1 by FUNCT_3:def 3
      .= 0;
    hence thesis by FUNCT_3:def 3;
  end;
  C = dom 1_minus(UMF(C)) & C = dom EMF(C) by FUNCT_2:def 1;
  hence thesis by A2,PARTFUN1:5;
end;
