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
  for f, g, h being Function
  st dom f = dom g & rng f c= dom h & rng g c= dom h &
  f, g are_fiberwise_equipotent holds h*f, h*g are_fiberwise_equipotent
proof
  let f, g, h be Function such that
A1: dom f = dom g and
A2: rng f c= dom h and
A3: rng g c= dom h and
A4: f, g are_fiberwise_equipotent;
  consider p being Permutation of dom f such that
A5: f = g*p by A1,A4,Th80;
A6: dom (h*f) = dom f by A2,RELAT_1:27;
A7: dom (h*g) = dom g by A3,RELAT_1:27;
  h*f = h*g*p by A5,RELAT_1:36;
  hence thesis by A1,A6,A7,Th80;
end;
