theorem Th55:
  r <> 0 implies Affin (r*AR) = r * Affin AR
 proof
  assume A1: r<>0;
  r"*(r*AR)=(r"*r)*AR by Th10
   .=1*AR by A1,XCMPLX_0:def 7
   .=AR by Th11;
  then AR c=r"*Affin(r*AR) by Lm7,Th9;
  then A2: Affin AR c=r"*Affin(r*AR) by Th51,Th54;
  r*AR c=r*Affin AR by Lm7,Th9;
  then A3: Affin(r*AR)c=r*Affin AR by Th51,Th54;
  r*(r"*Affin(r*AR))=(r*r")*Affin(r*AR) by Th10
   .=1*Affin(r*AR) by A1,XCMPLX_0:def 7
   .=Affin(r*AR) by Th11;
  then r*Affin AR c=Affin(r*AR) by A2,Th9;
  hence thesis by A3;
 end;
