reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem Th54:
  for C being Category st C is U-element holds
  the Source of C is Element of U & the Target of C is Element of U &
  the Comp of C is Element of U
  proof
    let C be Category;
    assume
A1: C is U-element;
    then reconsider cC = the carrier of C as Element of U;
    reconsider c9C = the carrier' of C as Element of U by A1;
    the Source of C c= [:c9C,cC:] & the Target of C c= [:c9C,cC:] &
      the Comp of C c= [:[:c9C,c9C:],c9C:];
    hence thesis by CLASSES4:13;
  end;
