
theorem
  for X,Y be non empty set, f be Function of [:X,Y:], ExtREAL holds
    ~(-f) = -(~f)
proof
   let X,Y be non empty set, f be Function of [:X,Y:],ExtREAL;
   now let z be Element of [:Y,X:];
    consider y,x be object such that
A1:  y in Y & x in X & z = [y,x] by ZFMISC_1:def 2;
A2: dom (-f) = [:X,Y:] & dom(-(~f)) = [:Y,X:] by FUNCT_2:def 1;
    reconsider y as Element of Y by A1;
    reconsider x as Element of X by A1;
    reconsider z1 = [x,y] as Element of [:X,Y:] by ZFMISC_1:87;
    ~(-f).z = ~(-f).(y,x) by A1; then
    ~(-f).z = (-f).(x,y) by FUNCT_4:def 8; then
    ~(-f).z = -(f.z1) by A2,MESFUNC1:def 7; then
    ~(-f).z = -(f.(x,y)); then
    ~(-f).z = -(~f.(y,x)) by FUNCT_4:def 8;
    hence ~(-f).z = (-(~f)).z by A1,A2,MESFUNC1:def 7;
   end;
   hence thesis by FUNCT_2:def 8;
end;
