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 Th52:
  parallelogram a,a9,b,b9 & parallelogram a,a9,c,c9 &
  parallelogram b,b9,d,d9 implies c,d // c9,d9
proof
  assume that
A1: parallelogram a,a9,b,b9 and
A2: parallelogram a,a9,c,c9 and
A3: parallelogram b,b9,d,d9;
A4: now
    assume
A5: not a,a9,d are_collinear;
    parallelogram b,b9,a,a9 by A1,Th43;
    then parallelogram a,a9,d,d9 by A3,A5,Th50;
    hence thesis by A2,Th49;
  end;
A6: now
A7: ( not a,a9,b are_collinear)& a,a9 // a,a9 by A1,Th1;
A8: a<>a9 by A1,Th36;
    assume that
A9: b,b9,c are_collinear and
A10: a,a9,d are_collinear;
    a<>b by A1,Th36;
    then consider x such that
A11: a,b,x are_collinear and
A12: x<>a and
A13: x<>b by Th48;
    a,b // a,x by A11;
    then consider y such that
A14: parallelogram a,a9,x,y by A12,A7,A8,Th23,Th44;
A15: not x,y,d are_collinear by A10,A14,Th39;
    parallelogram b,b9,x,y by A1,A11,A13,A14,Th51;
    then
A16: parallelogram x,y,d,d9 by A3,A15,Th50;
    not x,y,c are_collinear by A1,A9,A11,A13,A14,Th39,Th51;
    then parallelogram x,y,c,c9 by A2,A14,Th50;
    hence thesis by A16,Th49;
  end;
  now
    assume not b,b9,c are_collinear;
    then parallelogram b,b9,c,c9 by A1,A2,Th50;
    hence thesis by A3,Th49;
  end;
  hence thesis by A4,A6;
end;
