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 Th27:
  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 & a,c '||' c,a by PARSP_1:25;
  assume
A2: parallelogram a,b,c,d;
  then
A3: d<>b by Th26;
  a,c '||' b,d by A2;
  then
A4: a,c '||' d,b by PARSP_1:23;
  a,b '||' c,d by A2;
  then
A5: a,b '||' d,c by PARSP_1:23;
A6: ( not a,b,c are_collinear)& d<>c by A2,Th26;
A7: a,b '||' c,d & c <>a by A2,Th26;
  a,c '||' b,d & b<>a by A2,Th26;
  hence thesis by A1,A7,A5,A4,A6,A3,Th11;
end;
