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 Th42:
  parallelogram a,p,b,q & parallelogram a,p,c,r & parallelogram b,
  q,d,s implies c,d '||' r,s
proof
  assume that
A1: parallelogram a,p,b,q and
A2: parallelogram a,p,c,r and
A3: parallelogram b,q,d,s;
A4: now
    assume
A5: not a,p,d are_collinear;
    parallelogram b,q,a,p by A1,Th33;
    then parallelogram a,p,d,s by A3,A5,Th40;
    hence thesis by A2,Th39;
  end;
A6: now
A7: a<>p by A1,Th26;
A8: ( not a,p,b are_collinear)& a,p '||' a,p by A1,PARSP_1:25;
    assume that
A9: b,q,c are_collinear and
A10: a,p,d are_collinear;
    a<>b by A1,Th26;
    then consider x such that
A11: a,b,x are_collinear and
A12: x<>a and
A13: x<>b by Th38;
    a,b '||' a,x by A11;
    then consider y such that
A14: parallelogram a,p,x,y by A12,A8,A7,Th11,Th34;
A15: not x,y,d are_collinear by A10,A14,Th29;
    parallelogram b,q,x,y by A1,A11,A13,A14,Th41;
    then
A16: parallelogram x,y,d,s by A3,A15,Th40;
    not x,y,c are_collinear by A1,A9,A11,A13,A14,Th29,Th41;
    then parallelogram x,y,c,r by A2,A14,Th40;
    hence thesis by A16,Th39;
  end;
  now
    assume not b,q,c are_collinear;
    then parallelogram b,q,c,r by A1,A2,Th40;
    hence thesis by A3,Th39;
  end;
  hence thesis by A4,A6;
end;
