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;

theorem Th6:
  Gen x,y & Ortm(x,y,u)=Ortm(x,y,v) implies u=v
proof
  assume that
A1: Gen x,y and
A2: Ortm(x,y,u)=Ortm(x,y,v);
  pr1(x,y,u)*x + (-pr2(x,y,u))*y - (pr1(x,y,v)*x + (-pr2(x,y,v))*y)
  =0.V by A2,RLVECT_1:15;
  then pr1(x,y,u)*x + (-pr2(x,y,u))*y - (pr1(x,y,v)*x) - (-pr2(x,y,v))*y
  =0.V by RLVECT_1:27;
  then pr1(x,y,u)*x + (-(pr1(x,y,v))*x) + ((-pr2(x,y,u))*y) - (-pr2(x,y,v))*y
  =0.V by RLVECT_1:def 3;
  then pr1(x,y,u)*x - pr1(x,y,v)*x + ((-pr2(x,y,u))*y - (-pr2(x,y,v))*y)
  =0.V by RLVECT_1:def 3;
  then (pr1(x,y,u) - pr1(x,y,v))*x + ((-pr2(x,y,u))*y - (-pr2(x,y,v))*y)
  =0.V by RLVECT_1:35;
  then
A3: (pr1(x,y,u) - pr1(x,y,v))*x + ((-pr2(x,y,u)) - (-pr2(x,y,v)))*y
  =0.V by RLVECT_1:35;
  then
A4: pr1(x,y,u) - pr1(x,y,v)=0 by A1,ANALMETR:def 1;
  (-pr2(x,y,u)) - (-pr2(x,y,v))=0 by A1,A3,ANALMETR:def 1;
  hence u=pr1(x,y,v)*x + pr2(x,y,v)*y by A1,A4,Lm5
    .=v by A1,Lm5;
end;
