reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th3:
  ~([:X,Y:] --> z) = [:Y,X:] --> z
proof
A1: dom ([:X,Y:] --> z) = [:X,Y:];
  then
A2: dom ~([:X,Y:] --> z) = [:Y,X:] by FUNCT_4:46;
A3: now
    let x be object;
    assume
A4: x in [:Y,X:];
    then consider y1,y2 being object such that
A5: x = [y2,y1] and
A6: [y1,y2] in [:X,Y:] by A1,A2,FUNCT_4:def 2;
A7: ([:X,Y:] --> z).(y1,y2) = z by A6,FUNCOP_1:7;
    ([:Y,X:] --> z).(y2,y1) = z by A4,A5,FUNCOP_1:7;
    then (~([:X,Y:] --> z)).(y2,y1) = ([:Y,X:] --> z).(y2,y1) by A1,A6,A7,
FUNCT_4:def 2;
    hence (~([:X,Y:] --> z)).x = ([:Y,X:] --> z).x by A5;
  end;
  thus thesis by A2,A3;
end;
