theorem Th40:
  for POS being OrtAfSp for a,b,c being Element of POS, a9,b9,c9
being Element of the AffinStruct of POS st a=a9& b=b9 & c = c9
  holds (LIN a,b,c iff LIN a9,b9,c9)
proof
  let POS be OrtAfSp;
  let a,b,c be Element of POS, a9,b9,c9 be Element of the AffinStruct of POS
   such that
A1: a=a9 & b=b9 & c = c9;
  hereby
    assume LIN a,b,c;
    then a,b // a,c;
    then a9,b9 // a9,c9 by A1,Th36;
    hence LIN a9,b9,c9 by AFF_1:def 1;
  end;
  assume LIN a9,b9,c9;
  then a9,b9 // a9,c9 by AFF_1:def 1;
  then a,b // a,c by A1,Th36;
  hence thesis;
end;
