reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);
reserve x,a,b,c,d,p,q,y for Element of POS;
reserve A,K,M for Subset of POS;
reserve POS for OrtAfSp;
reserve A,K,M,N for Subset of POS;
reserve a,b,c,d,p,q,r,s for Element of POS;
reserve POS for OrtAfPl;
reserve K,M,N for Subset of POS;
reserve x,a,b,c,d,p,q for Element of POS;

theorem Th63:
  p<>q & ( p,q _|_ a,b & p,q _|_ c,d or p,q _|_ a,b & c,d _|_ p,q
or a,b _|_ p,q & c,d _|_ p,q or a,b _|_ p,q & p,q _|_ c,d ) implies a,b // c,d
proof
  assume that
A1: p<>q and
A2: p,q _|_ a,b & p,q _|_ c,d or p,q _|_ a,b & c,d _|_ p,q or a,b _|_ p,
  q & c,d _|_ p,q or a,b _|_ p,q & p,q _|_ c,d;
  p,q _|_ a,b & p,q _|_ c,d by A2,Th61;
  hence thesis by A1,Def8;
end;
