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 Th32:
  X c= dom f & f is one-to-one implies X,f.:X are_equipotent
proof
  assume that
A1: X c= dom f and
A2: f is one-to-one;
  take g = f|X;
  thus g is one-to-one by A2,FUNCT_1:52;
  thus dom g = X by A1,RELAT_1:62;
  thus thesis by RELAT_1:115;
end;
