reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem Th32:
  K // M implies a*K=a*M
proof
  assume
A1: K // M;
  then
A2: K is being_line by AFF_1:36;
  then K // a*K by Def3;
  then
A3: a*K // M by A1,AFF_1:44;
A4: M is being_line by A1,AFF_1:36;
  then
A5: a in a*M by Def3;
  M // a*M by A4,Def3;
  then
A6: a*K // a*M by A3,AFF_1:44;
  a in a*K by A2,Def3;
  hence thesis by A5,A6,AFF_1:45;
end;
