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
 for f being Function-yielding Function
  st x in dom f & f.x is Function & y in dom <:f:>
 holds f..(x,y) = <:f:>..( y, x )
proof let f be Function-yielding Function;
  assume that
A1: x in dom f and
 f.x is Function and
A2: y in dom <:f:>;
  reconsider g = f.x, h = <:f:>.y as Function;
A3: dom h = dom f by A2,Th26;
A4: g in rng f by A1,FUNCT_1:def 3;
A5: x in dom h by A1,A3;
  y in dom g by A2,A4,Th27;
  hence f..(x,y) = g.y by A1,FUNCT_5:38
    .= h.x by A1,A2,Th29
    .= <:f:>..(y,x) by A2,A5,FUNCT_5:38;
end;
