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 Th59:
  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
  reconsider a9=a,b9=b,c9= c,d9=d as Element of the AffinStruct of POS;
  assume a,b // c,d;
  then
A1: a9,b9 // c9,d9 by Th36;
  then
A2: b9,a9 // d9,c9 & c9,d9 // a9,b9 by AFF_1:4;
A3: d9,c9 // b9,a9 by A1,AFF_1:4;
A4: c9,d9 // b9,a9 & d9,c9 // a9,b9 by A1,AFF_1:4;
  a9,b9 // d9,c9 & b9,a9 // c9,d9 by A1,AFF_1:4;
  hence thesis by A2,A4,A3,Th36;
end;
