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 Th61:
  congr a,b,c,d & a,b,c are_collinear & parallelogram r,s,a,b
  implies parallelogram r,s,c,d
proof
  assume that
A1: congr a,b,c,d and
A2: a,b,c are_collinear and
A3: parallelogram r,s,a,b;
  now
A4: parallelogram a,b,r,s by A3,Th43;
    assume
A5: a<>b;
    then consider p,q such that
A6: parallelogram p,q,a,b and
A7: parallelogram p, q,c,d by A1;
    parallelogram a,b,p,q by A6,Th43;
    then
A8: r,c // s,d by A7,A4,Th52;
    r,s // a,b & a,b // c,d by A1,A3,Th57;
    then
A9: r,s // c,d by A5,Th8;
    not r,s,c are_collinear by A2,A3,Th39;
    hence thesis by A9,A8;
  end;
  hence thesis by A3,Th36;
end;
