theorem Th53:
  Affin (v+A) = v + Affin A
 proof
  v+A c=v+Affin A by Lm7,RLTOPSP1:8;
  then A1: Affin(v+A)c=v+Affin A by Th51,RUSUB_4:31;
  -v+(v+A)=(-v+v)+A by Th5
   .=0.V+A by RLVECT_1:5
   .=A by Th6;
  then A c=-v+Affin(v+A) by Lm7,RLTOPSP1:8;
  then A2: Affin A c=-v+Affin(v+A) by Th51,RUSUB_4:31;
  v+(-v+Affin(v+A))=(v+-v)+Affin(v+A) by Th5
   .=0.V+Affin(v+A) by RLVECT_1:5
   .=Affin(v+A) by Th6;
  then v+Affin A c=Affin(v+A) by A2,RLTOPSP1:8;
  hence thesis by A1;
 end;
