
theorem Th17:
  for X,Y be non empty set,
      S be SigmaField of X, T be Function of X,Y
   st T is bijective holds
    ex H be Function of Y,X
     st H is bijective
     & H = T" & H" = T & .:H = (.:T)" & (.:H).:CopyField(T,S) = S
     & CopyField(H,CopyField(T,S)) = S
proof
    let X,Y be non empty set, S be SigmaField of X, T be Function of X,Y;
    assume
A1: T is bijective; then
    .:T is bijective by Th1; then
A2: dom (.:T) = bool X & rng (.:T) = bool Y
      by FUNCT_2:def 1,def 3;

    consider H be Function of Y,X such that
A3: H is bijective & H =T" & .:H = (.:T)" by A1,Th14;
    take H;
    thus H is bijective & H = T" & H" = T & .:H = (.:T)" by A1,A3,FUNCT_1:43;
    (.:H).:CopyField(T,S) = (.:H).:((.:T).:S) by A1,Def2; then
    (.:H).:CopyField(T,S) = (.:T)"((.:T).:S) by A1,A3,FUNCT_1:85;
    hence (.:H).:CopyField(T,S) = S by A1,A2,FUNCT_1:94;
    hence CopyField(H,CopyField(T,S)) = S by A3,Def2;
end;
