
theorem Th10:
  for V being RealLinearSpace holds for o,u,v,u1,v1 being VECTOR
of V, r being Real
   st o-u=r*(u1-o) & r<>0 & o,v '||' o,v1 & not o,u '||' o,v &
u,v '||' u1,v1 holds v1 = u1 + (-r)"*(v-u) & v1 = o + (-r)"*(v-o) & v-u = (-r)*
  (v1-u1)
proof
  let V be RealLinearSpace;
  let o,u,v,u1,v1 be VECTOR of V, r be Real such that
A1: o-u=r*(u1-o) and
A2: r<>0 and
A3: o,v '||' o,v1 and
A4: not o,u '||' o,v and
A5: u,v '||' u1,v1;
A6: -r <> 0 by A2;
  for r1,r2 being Real st r1*(u-o)+r2*(v-o) = 0.V holds r1=0 & r2=0
  proof
    let r1,r2 be Real;
    assume r1*(u-o)+r2*(v-o) = 0.V;
    then
A7: r1*(u-o) = -(r2*(v-o)) by RLVECT_1:6
      .= r2*(-(v-o)) by RLVECT_1:25
      .= (-r2)*(v-o) by RLVECT_1:24;
    assume r1<>0 or r2<>0;
    then r1<>0 or -r2<>0;
    then o,u // o,v or o,u // v,o by A7,ANALMETR:14;
    hence contradiction by A4,GEOMTRAP:def 1;
  end;
  then reconsider X = OASpace(V) as OAffinSpace by ANALOAF:26;
  set w = u1 + (-r)"*(v-u);
  reconsider p=o,a=u,a1=u1,b=v,b1=v1,q=w as Element of X by Th3;
  a,b '||' a1,b1 by A5,Th4;
  then
A8: b,a '||' a1,b1 by DIRAF:22;
  p,b '||' p,b1 by A3,Th4;
  then
A9: p,b,b1 are_collinear by DIRAF:def 5;
A10: (-r)*(w-u1) = (-r)*((-r)"*(v-u)) by RLSUB_2:61
    .= ((-r)*(-r)")*(v-u) by RLVECT_1:def 7
    .= 1*(v-u) by A6,XCMPLX_0:def 7;
  then
A11: v-u = (-r)*(w-u1) by RLVECT_1:def 8;
  u,v // u1,w or u,v // w,u1 by A10,ANALMETR:14;
  then u,v '||' u1,w by GEOMTRAP:def 1;
  then a,b '||' a1,q by Th4;
  then
A12: b,a '||' a1,q by DIRAF:22;
A13: (-r)*(o-w) = (-r)*o - (-r)*w by RLVECT_1:34
    .= (-r)*o - ((-r)*u1 + (-r)*((-r)"*(v-u))) by RLVECT_1:def 5
    .= (-r)*o - ((-r)*u1 + ((-r)*(-r)")*(v-u)) by RLVECT_1:def 7
    .= (-r)*o - ((-r)*u1 + 1*(v-u)) by A6,XCMPLX_0:def 7
    .= (-r)*o - ((-r)*u1 + (v-u)) by RLVECT_1:def 8
    .= ((-r)*o - (-r)*u1) - (v-u) by RLVECT_1:27
    .= (-r)*(o-u1) - (v-u) by RLVECT_1:34
    .= r*(-(o-u1)) - (v-u) by RLVECT_1:24
    .= (o-u) - (v-u) by A1,RLVECT_1:33
    .= o - ((v-u)+u) by RLVECT_1:27
    .= o - v by RLSUB_2:61
    .= 1*(o-v) by RLVECT_1:def 8;
  then v,o // w,o or v,o // o,w by ANALMETR:14;
  then o,v // w,o or o,v // o,w by ANALOAF:12;
  then o,v '||' o,w by GEOMTRAP:def 1;
  then p,b '||' p,q by Th4;
  then
A14: p,b,q are_collinear by DIRAF:def 5;
  1*(u-o) = (-1)*(-(u-o)) by RLVECT_1:26
    .= (-1)*(r*(u1-o)) by A1,RLVECT_1:33
    .= ((-1)*r)*(u1-o) by RLVECT_1:def 7
    .= (-r)*(u1-o);
  then o,u // o,u1 or o,u // u1,o by ANALMETR:14;
  then o,u '||' o,u1 by GEOMTRAP:def 1;
  then p,a '||' p,a1 by Th4;
  then
A15: p,a,a1 are_collinear by DIRAF:def 5;
A16: not p,b,a are_collinear
  proof
    assume p,b,a are_collinear;
    then p,b '||' p,a by DIRAF:def 5;
    then p,a '||' p,b by DIRAF:22;
    hence contradiction by A4,Th4;
  end;
A17: (-r)*(w-o) = r*(-(w-o)) by RLVECT_1:24
    .= r*(o-w) by RLVECT_1:33
    .= -(-(r*(o-w))) by RLVECT_1:17
    .= -(r*(-(o-w))) by RLVECT_1:25
    .= -(1*(o-v)) by A13,RLVECT_1:24
    .= -(o-v) by RLVECT_1:def 8
    .= v-o by RLVECT_1:33;
  w = o + (w-o) by RLSUB_2:61
    .= o + (-r)"*(v-o) by A6,A17,ANALOAF:6;
  hence thesis by A11,A16,A9,A14,A15,A12,A8,PASCH:4;
end;
