reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;
reserve n for Element of omega;
reserve e,u for set;

theorem Th80:
  for F,G be Function st dom F = dom G holds
  F,G are_fiberwise_equipotent iff ex P be Permutation of dom F st F = G*P
proof
  let F,G be Function;
  assume
A1: dom F = dom G;
  thus F,G are_fiberwise_equipotent implies
  ex P be Permutation of dom F st F = G*P
  proof
    assume F,G are_fiberwise_equipotent;
    then consider I be Function such that
A2: dom I = dom F and
A3: rng I = dom G and
A4: I is one-to-one and
A5: F = G*I by Th77;
    reconsider I as Function of dom F,dom F by A1,A2,A3,FUNCT_2:2;
    reconsider I as Permutation of dom F by A1,A3,A4,FUNCT_2:57;
    take I;
    thus thesis by A5;
  end;
  given P be Permutation of dom F such that
A6: F = G*P;
 P is Function of dom F,dom F & dom F = {} implies dom F ={};
  then rng P = dom F & dom P = dom F by FUNCT_2:def 1,def 3;
  hence thesis by A1,A6,Th77;
end;
