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 Th61:
  a,b _|_ c,d implies a,b _|_ d,c & b,a _|_ c,d & b,a _|_ d,c & c,
  d _|_ a,b & c,d _|_ b,a & d,c _|_ a,b & d,c _|_ b,a
proof
  assume
A1: a,b _|_ c,d;
  then a,b _|_ d,c by Def7;
  then
A2: d,c _|_ a,b by Def7;
  then d,c _|_ b,a by Def7;
  then b,a _|_ d,c by Def7;
  then b,a _|_ c,d by Def7;
  hence thesis by A1,A2,Def7;
end;
