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;

theorem Th62:
  p<>q & ( p,q // a,b & p,q _|_ c,d or p,q // c,d & p,q _|_ a,b or
p,q // a,b & c,d _|_ p,q or p,q // c,d & a,b _|_ p,q or a,b // p,q & c,d _|_ p,
  q or c,d // p,q & a,b _|_ p,q or a,b // p,q & p,q _|_ c,d or c,d // p,q & p,q
  _|_ a,b ) implies a,b _|_ c,d
proof
  assume that
A1: p<>q and
A2: p,q // a,b & p,q _|_ c,d or p,q // c,d & p,q _|_ a,b or p,q // a,b &
c,d _|_ p,q or p,q // c,d & a,b _|_ p,q or a,b // p,q & c,d _|_ p,q or c,d // p
  ,q & a,b _|_ p,q or a,b // p,q & p,q _|_ c,d or c,d // p,q & p,q _|_ a,b;
A3: now
    assume 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;
    then p,q // a,b & p,q _|_ c,d by Th59,Th61;
    then c,d _|_ a,b by A1,Def7;
    hence thesis by Th61;
  end;
  now
    assume p,q // c,d & p,q _|_ a,b or p,q // c,d & a,b _|_ p,q or c,d // p,
    q & a,b _|_ p,q or c,d // p,q & p,q _|_ a,b;
    then p,q // c,d & p,q _|_ a,b by Th59,Th61;
    hence thesis by A1,Def7;
  end;
  hence thesis by A2,A3;
end;
