reserve X,X1,X2,Y,Y1,Y2 for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,g1,g2,h for Function,
  R,S for Relation;

theorem Th81:
  f is one-to-one implies f"(f.:X) c= X
proof
  assume
A1: f is one-to-one;
  let x be object;
  assume
A2: x in f"(f.:X);
  then f.x in f.:X by Def7;
  then
A3: ex z being object st z in dom f & z in X & f.x = f.z by Def6;
  x in dom f by A2,Def7;
  hence thesis by A1,A3;
end;
