theorem Th45:
  {f,g}++{h,i} = {f+h,f+i,g+h,g+i}
proof
  thus {f,g}++{h,i} = ({f}\/{g})++{h,i} by ENUMSET1:1
    .= ({f}++{h,i}) \/ ({g}++{h,i}) by Th41
    .= {f+h,f+i} \/ ({g}++{h,i}) by Th44
    .= {f+h,f+i} \/ {g+h,g+i} by Th44
    .= {f+h,f+i,g+h,g+i} by ENUMSET1:5;
end;
