theorem Th54:
  AR is Affine implies r * AR is Affine
 proof
  assume A1: AR is Affine;
  let v1,v2 be VECTOR of R,s;
  assume v1 in r*AR;
  then consider w1 be Element of R such that
   A2: v1=r*w1 and
   A3: w1 in AR;
  assume v2 in r*AR;
  then consider w2 be Element of R such that
   A4: v2=r*w2 and
   A5: w2 in AR;
  A6: (1-s)*w1+s*w2 in AR by A1,A3,A5;
  A7: (1-s)*(r*w1)=((1-s)*r)*w1 by RLVECT_1:def 7
   .=r*((1-s)*w1) by RLVECT_1:def 7;
  s*(r*w2)=(s*r)*w2 by RLVECT_1:def 7
   .=r*(s*w2) by RLVECT_1:def 7;
  then (1-s)*v1+s*v2=r*((1-s)*w1+s*w2) by A2,A4,A7,RLVECT_1:def 5;
  hence thesis by A6;
 end;
