reserve V for RealLinearSpace,
  u,u1,u2,v,v1,v2,w,w1,x,y for VECTOR of V,
  a,a1,a2,b,b1,b2,c1,c2,n,k1,k2 for Real;
reserve uu,vv for object;
reserve p,q,r,s for Element of CESpace(V,x,y);
reserve p,q,r,s for Element of CMSpace(V,x,y);

theorem
  u=p & v=q & u1=r & v1=s implies
   (p,q _|_ r,s iff u,v,u1,v1 are_COrtm_wrt x,y )
proof
  assume that
A1: u=p and
A2: v=q and
A3: u1=r and
A4: v1=s;
A5: p,q _|_ r,s implies u,v,u1,v1 are_COrtm_wrt x,y
  proof
    assume p,q _|_ r,s;
    then [[p,q],[r,s]] in the orthogonality of CMSpace(V,x,y)
       by ANALMETR:def 5;
    then consider u19,u29,v19,v29 being VECTOR of V such that
A6: [u,v]=[u19,u29] and
A7: [u1,v1]=[v19,v29] and
A8: u19,u29,v19,v29 are_COrtm_wrt x,y by A1,A2,A3,A4,Def6;
A9: u=u19 by A6,XTUPLE_0:1;
A10: v=u29 by A6,XTUPLE_0:1;
    u1=v19 by A7,XTUPLE_0:1;
    hence thesis by A7,A8,A9,A10,XTUPLE_0:1;
  end;
  u,v,u1,v1 are_COrtm_wrt x,y implies p,q _|_ r,s
  proof
    assume u,v,u1,v1 are_COrtm_wrt x,y;
    then [[u,v],[u1,v1]] in the orthogonality of
     OrtStr (# the carrier of V,CORTM(V,x,y) #) by Def6;
    hence thesis by A1,A2,A3,A4,ANALMETR:def 5;
  end;
  hence thesis by A5;
end;
