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
  a<>d implies ex b,c st parallelogram a,b,c,d
proof
  assume a<>d;
  then consider b such that
A1: not a,d,b are_collinear by Th14;
  not b,a,d are_collinear by A1,Th10;
  then consider c such that
A2: parallelogram b,a,d,c by Th34;
  parallelogram a,b,c,d by A2,Th33;
  hence thesis;
end;
