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);

theorem Th21:
  p=u & p1=u1 & q=v & q1=v1 implies (p,q _|_ p1,q1 iff u,v,u1,v1
  are_Ort_wrt w,y)
proof
  assume
A1: p=u & p1=u1 & q=v & q1=v1;
  hereby
    assume p,q _|_ p1,q1;
    then consider u9,v9,u19,v19 being VECTOR of V such that
A2: [u,v] = [u9,v9] and
A3: [u1,v1] = [u19,v19] and
A4: u9,v9,u19,v19 are_Ort_wrt w,y by A1,Def4;
A5: u1=u19 by A3,XTUPLE_0:1;
    u=u9 & v=v9 by A2,XTUPLE_0:1;
    hence u,v,u1,v1 are_Ort_wrt w,y by A3,A4,A5,XTUPLE_0:1;
  end;
  assume u,v,u1,v1 are_Ort_wrt w,y;
  hence thesis by A1,Def4;
end;
