reserve MS for OrtAfPl;
reserve MP for OrtAfSp;
reserve V for RealLinearSpace;
reserve w,y,u,v for VECTOR of V;

theorem Th18:
  for a,b being Real st Gen w,y & 0.V<>u & 0.V<>v & u,v
  are_Ort_wrt w,y & u=a*w+b*y
   ex c being Real st c <>0 & v=(c*b)*w+(-c*a)*y
proof
  let a,b be Real such that
A1: Gen w,y and
A2: 0.V<>u and
A3: 0.V<>v and
A4: u,v are_Ort_wrt w,y and
A5: u=a*w+b*y;
  set v9=b*w+(-a)*y;
  v9= (1*b)*w+(-1*a)*y;
  then u,v9 are_Ort_wrt w,y by A1,A5,Lm5;
  then consider a1,b1 being Real such that
A6: a1*v = b1*v9 and
A7: a1<>0 or b1<>0 by A1,A2,A4,ANALMETR:9;
A8: now
    assume
A9: a1=0;
    then 0.V = b1*v9 by A6,RLVECT_1:10;
    then v9=0.V by A7,A9,RLVECT_1:11;
    then b=0 & -a=0 by A1,ANALMETR:def 1;
    then u= 0.V + 0*y by A5,RLVECT_1:10
      .= 0.V + 0.V by RLVECT_1:10
      .= 0.V by RLVECT_1:4;
    hence contradiction by A2;
  end;
  take c =a1"*b1;
A10: now
    assume
A11: b1=0;
    then 0.V = a1*v by A6,RLVECT_1:10;
    hence contradiction by A3,A7,A11,RLVECT_1:11;
  end;
  now
    assume c =0;
    then a1"=0 by A10,XCMPLX_1:6;
    hence contradiction by A8,XCMPLX_1:202;
  end;
  hence c <>0;
  thus v=(a1")*(b1*v9) by A6,A8,ANALOAF:5
    .= c*v9 by RLVECT_1:def 7
    .= c*(b*w) + c*((-a)*y) by RLVECT_1:def 5
    .= (c*b)*w + c*((-a)*y) by RLVECT_1:def 7
    .= (c*b)*w + (c*(-a))*y by RLVECT_1:def 7
    .= (c*b)*w + (-c*a)*y;
end;
