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 Th39:
  parallelogram a,p,b,q & parallelogram a,p,c,r implies b,c '||' q ,r
proof
  assume that
A1: parallelogram a,p,b,q and
A2: parallelogram a,p,c,r;
A3: a,p '||' c,r & a,c '||' p,r by A2;
  not a,p,c are_collinear by A2;
  then
A4: not a,p '||' a,c;
  not a,p,b are_collinear by A1;
  then
A5: not a,p '||' a,b;
  a,p '||' b,q & a,b '||' p,q by A1;
  hence thesis by A5,A4,A3,Def1;
end;
