reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;
reserve S for Sequence;

theorem Th37:
  X,Y are_equipotent & X is finite implies Y is finite
proof
  assume X,Y are_equipotent;
  then consider f such that
  f is one-to-one and
A1: dom f = X and
A2: rng f = Y;
  given p being Function such that
A3: rng p = X and
A4: dom p in omega;
  take f*p;
  thus rng(f*p) = Y by A1,A2,A3,RELAT_1:28;
  thus thesis by A1,A3,A4,RELAT_1:27;
end;
