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 Th37:
  parallelogram a,b,c,d implies not a,b,c are_collinear & not b,a,d
  are_collinear & not c,d,a are_collinear & not d,c,b are_collinear
proof
A1: a,b // b,a by Def1;
  assume
A2: parallelogram a,b,c,d;
  hence not a,b,c are_collinear;
A3: b<>a & b<>d by A2,Th36;
  a,c // b,d by A2;
  then
A4: a,c // d,b by Th6;
  a,b // c,d by A2;
  then
A5: a,b // d,c by Th6;
  ( not a,b,c are_collinear)& a,c // b,d by A2;
  hence not b,a,d are_collinear by A1,A3,Th23;
A6: a,c // c,a by Def1;
A7: c <>d & c <>a by A2,Th36;
  ( not a,b,c are_collinear)& a,b // c,d by A2;
  hence not c,d,a are_collinear by A6,A7,Th23;
A8: d<>b by A2,Th36;
  ( not a,b,c are_collinear)& c <>d by A2,Th36;
  hence thesis by A5,A4,A8,Th23;
end;
