reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem
  trap a,p,b,q,o & trap a,p,c,r,o & trap b,q,d,s,o implies c,d // r,s
proof
  assume that
A1: trap a,p,b,q,o and
A2: trap a,p,c,r,o and
A3: trap b,q,d,s,o;
A4: now
    assume
A5: not o,a,d are_collinear;
    trap b,q,a,p,o by A1,Th93;
    then trap a,p,d,s,o by A3,A5,Th99;
    hence thesis by A2,Th98;
  end;
A6: now
    not o,a,b are_collinear by A1;
    then not b,a,o are_collinear by Th22;
    then consider x such that
A7: parallelogram b,a,o,x by Th44;
    o,b,q are_collinear by A1;
    then o,b // o,q;
    then
A8: b,o // q,o by Th6;
A9: o,x // o,x by Th1;
    b,o // a,x & b<>o by A7,Th36;
    then
A10: q,o // a,x by A8,Def1;
A11: not o,x,b are_collinear by A7,Th38;
A12: o<>d by A3,Th90;
    assume that
A13: o<>p and
A14: o,b,c are_collinear and
A15: o,a,d are_collinear;
    not o,p,q are_collinear by A1,A13,Th95;
    then not q,p,o are_collinear by Th22;
    then consider y such that
A16: parallelogram q,p,o,y by Th44;
A17: not o,x,a are_collinear by A7,Th38;
A18: o<>c by A2,Th90;
    a,b // p,q by A1;
    then
A19: b,a // q,p by Th6;
A20: o,a,p are_collinear by A1;
    b,a // o,x & b<>a by A7,Th36;
    then
A21: q,p // o,x by A19,Def1;
A22: o<>x by A7,Th36;
    q<>p & q,p // o,y by A16,Th36;
    then o,x // o,y by A21,Def1;
    then
A23: o,x,y are_collinear;
    q,o // p,y & q<>o by A16,Th36;
    then
A24: a,x // p,y by A10,Def1;
    not o,a,x are_collinear by A7,Th38;
    then
A25: trap a,p,x,y,o by A23,A24,A20;
    not o,b,x are_collinear by A7,Th38;
    then
A26: trap b,q,x,y,o by A1,A25,Th99;
    o,a // o,d by A15;
    then
A27: trap x,y,d,s,o by A3,A26,A22,A12,A17,A9,Th23,Th99;
    o,b // o,c by A14;
    then trap x,y,c,r,o by A2,A25,A11,A22,A18,A9,Th23,Th99;
    hence thesis by A27,Th98;
  end;
A28: now
    assume
A29: o=p;
    then o=q by A1,Th93,Th94;
    then
A30: o=s by A3,Th93,Th94;
    o=r by A2,A29,Th93,Th94;
    hence thesis by A30,Def1;
  end;
  now
    assume not o,b,c are_collinear;
    then trap b,q,c,r,o by A1,A2,Th99;
    hence thesis by A3,Th98;
  end;
  hence thesis by A4,A28,A6;
end;
