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;
