theorem Th31:
  A is being_line implies a*A = a*(q*A)
proof
  assume
A1: A is being_line;
  then A // q*A & A // a*A by Def3;
  then
A2: a*A // q*A by AFF_1:44;
A3: q*A is being_line by A1,Th27;
  then
A4: a in a*(q*A) by Def3;
  q*A // a*(q*A) by A3,Def3;
  then
A5: a*A // a*(q*A) by A2,AFF_1:44;
  a in a*A by A1,Def3;
  hence thesis by A4,A5,AFF_1:45;
end;
