theorem Th16:
  a,b '||' a,c implies b,a '||' b,c
proof
  assume a,b '||' a,c;
  then consider e,f,g,h such that
A1: [[e,f],[g,h]] = [[a,b],[a,c]] and
A2: (e`1_3-f`1_3)*(g`2_3-h`2_3) - (g`1_3-h`1_3)*(e`2_3-f`2_3) = 0.F &
  (e`1_3-f`1_3)*(g`3_3-h`3_3
  ) - (g`1_3 -h`1_3)*(e`3_3-f`3_3) = 0.F and
A3: (e`2_3-f`2_3)*(g`3_3-h`3_3) - (g`2_3-h`2_3)*(e`3_3-f`3_3) = 0.F by Th12;
A4: e = a by A1,MCART_1:93;
A5: g = a by A1,MCART_1:93;
  then
A6: (f`2_3-e`2_3)*(f`3_3-h`3_3) - (f`2_3-h`2_3)*(f`3_3-e`3_3) = 0.F
   by A3,A4,Lm13;
  f = b by A1,MCART_1:93;
  then
A7: [[f,e],[f,h]] = [[b,a],[b,c]] by A1,A4,MCART_1:93;
  (f`1_3-e`1_3)*(f`2_3-h`2_3) - (f`1_3-h`1_3)*(f`2_3-e`2_3) = 0.F &
   (f`1_3-e`1_3)*(f`3_3-h`3_3) -
  (f`1_3 -h`1_3)*(f`3_3-e`3_3) = 0.F by A2,A4,A5,Lm13;
  hence thesis by A7,A6,Th12;
end;
