reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem Th51:
  a,b,c are_collinear & b<>c & parallelogram a,a9,b,b9 &
  parallelogram a,a9,c,c9 implies parallelogram b,b9,c,c9
proof
  assume that
A1: a,b,c are_collinear and
A2: b<>c and
A3: parallelogram a,a9,b,b9 and
A4: parallelogram a,a9,c,c9;
A5: b<>b9 by A3,Th36;
  a,b // a,c by A1;
  then
A6: a,b // b,c by Th7;
  thus thesis by A2,A3,A4,A6,A5,Th23,Th50;
end;
