reserve F for Field;
reserve a,b,c,d,p,q,r for Element of MPS(F);
reserve e,f,g,h,i,j,k,l,m,n,o,w for Element of [:the carrier of F,the carrier
  of F,the carrier of F:];
reserve K,L,M,N,R,S for Element of F;
reserve FdSp for FanodesSp;
reserve a,b,c,d,p,q,r,s,o,x,y for Element of FdSp;

theorem Th28:
  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 & not a,c,b
  are_collinear & not b,a,c are_collinear & not b,c,a are_collinear & not c,a,b
  are_collinear & not c,b,a are_collinear & not b,d,a are_collinear & not a,b,d
  are_collinear & not a,d,b are_collinear & not d,a,b are_collinear & not d,b,a
  are_collinear & not c,a,d are_collinear & not a,c,d are_collinear & not a,d,c
  are_collinear & not d,a,c are_collinear & not d,c,a are_collinear & not d,b,c
  are_collinear & not b,c,d are_collinear & not b,d,c are_collinear & not c,b,d
  are_collinear & not c,d,b are_collinear
proof
  assume
A1: parallelogram a,b,c,d;
  then
A2: ( not c,d,a are_collinear)& not d,c,b are_collinear by Th27;
  ( not a,b,c are_collinear)& not b,a,d are_collinear by A1,Th27;
  hence thesis by A2,Th10;
end;
