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;

theorem Th6:
  u,v are_Ort_wrt w,y implies a*u,b*v are_Ort_wrt w,y
proof
  assume u,v are_Ort_wrt w,y;
  then consider a1,a2,b1,b2 such that
A1: u = a1*w + a2*y and
A2: v = b1*w + b2*y and
A3: a1*b1 + a2*b2 = 0;
A4: b*v = b*(b1*w) + b*(b2*y) by A2,RLVECT_1:def 5
    .= (b*b1)*w + b*(b2*y) by RLVECT_1:def 7
    .= (b*b1)*w + (b*b2)*y by RLVECT_1:def 7;
A5: (a*a1)*(b*b1) + (a*a2)*(b*b2) = b*a*(a1*b1 + a2*b2) .= 0 by A3;
  a*u = a*(a1*w) + a*(a2*y) by A1,RLVECT_1:def 5
    .= (a*a1)*w + a*(a2*y) by RLVECT_1:def 7
    .= (a*a1)*w + (a*a2)*y by RLVECT_1:def 7;
  hence thesis by A4,A5;
end;
