reserve T1,T2,T3 for TopSpace,
  A1 for Subset of T1, A2 for Subset of T2, A3 for Subset of T3;
reserve n,k for Nat;
reserve M,N for non empty TopSpace;
reserve p,q,p1,p2 for Point of TOP-REAL n;
reserve r for Real;

theorem Th26:
  p <> 0.TOP-REAL n implies
  ex A being linearly-independent Subset of TOP-REAL n
  st card A = n - 1 & [#]Lin(A) = Plane(p,0.TOP-REAL n)
proof
  assume
A1: p <> 0.TOP-REAL n;
  reconsider n1 = n as Element of NAT by ORDINAL1:def 12;
A2: 0.TOP-REAL n = -0.TOP-REAL n1 by JGRAPH_5:10
  .= -0.TOP-REAL n;
  set V1 = Plane(p,0.TOP-REAL n);
  |( p,0.TOP-REAL n )| = 0 by EUCLID_2:32;
  then |( p,0.TOP-REAL n-0.TOP-REAL n )| = 0 by RLVECT_1:5;
  then
A3: 0.TOP-REAL n in V1;
A4: for v, u being VECTOR of TOP-REAL n st v in V1 & u in V1 holds v + u in V1
  proof
    let v, u be VECTOR of TOP-REAL n;
    assume v in V1;
    then consider v1 be Point of TOP-REAL n such that
A5: v = v1 & |( p,v1-0.TOP-REAL n )| = 0;
    assume u in V1;
    then consider u1 be Point of TOP-REAL n such that
A6: u = u1 & |( p,u1-0.TOP-REAL n )| = 0;
    |( p,(v1+u1)-0.TOP-REAL n )| = |( p,v1+(u1-0.TOP-REAL n) )| by
RLVECT_1:def 3
    .= |( p, v1 )| + |( p, u1-0.TOP-REAL n )| by EUCLID_2:26
    .= 0 by A5,A2,A6,RLVECT_1:4;
    hence v + u in V1 by A5,A6;
  end;
  for a being Real for v being VECTOR of TOP-REAL n st v in V1
  holds a * v in V1
  proof
    let a be Real;
    let v be VECTOR of TOP-REAL n;
    assume v in V1;
    then consider v1 be Point of TOP-REAL n such that
A7: v = v1 & |( p,v1-0.TOP-REAL n )| = 0;
    |( p,a*(v1-0.TOP-REAL n) )| = a * (0 qua Real) by A7,EUCLID_2:20;
    then |( p,a*v1 - a*0.TOP-REAL n )| = 0 by RLVECT_1:34;
    then |( p, a*v1 - 0.TOP-REAL n )| = 0 by RLVECT_1:10;
    hence a * v in V1 by A7;
  end;
  then consider W be strict Subspace of TOP-REAL n such that
A8: V1 = the carrier of W by A3,RLSUB_1:35,A4,RLSUB_1:def 1;
  set A1 = the Basis of W;
A10: [#]Lin A1 = Plane(p,0.TOP-REAL n) by A8,RLVECT_3:def 3;
  reconsider A = A1 as linearly-independent Subset of TOP-REAL n
  by RLVECT_3:def 3,RLVECT_5:14;
  take A;
  reconsider A2 = {p} as linearly-independent Subset of TOP-REAL n
  by A1,RLVECT_3:8;
A11: dim Lin A2 = card A2 by RLVECT_5:29 .= 1 by CARD_1:30;
  for v being VECTOR of TOP-REAL n ex v1,v2 being VECTOR of TOP-REAL n
  st v1 in Lin A & v2 in Lin A2 & v = v1 + v2
  proof
    let v be VECTOR of TOP-REAL n;
    set a = |( p,v )| / |( p,p )|;
    set v2 = a * p;
    set v1 = v - v2;
    reconsider v1,v2 as VECTOR of TOP-REAL n;
    take v1,v2;
A12: |( p,p )| > 0 by A1,EUCLID_2:43;
    |( p,v1 )| = |( p, v )| - |( p, v2 )| by EUCLID_2:27
    .= |( p, v )| - a * |( p, p )| by EUCLID_2:20
    .= |( p, v )| - |( p, v )| by A12,XCMPLX_1:87
    .= 0;
    then |( p,v1 - 0.TOP-REAL n )| = 0 by A2,RLVECT_1:4;
    then v1 in Lin A1 by A10;
    hence v1 in Lin A by RLVECT_5:20;
    thus v2 in Lin A2 by RLVECT_4:8;
    thus thesis by RLVECT_4:1;
  end;
  then
A13: the RLSStruct of TOP-REAL n = Lin(A) + Lin(A2) by RLSUB_2:44;
  Lin(A) /\ Lin(A2) = (0).TOP-REAL n
  proof
    for v being VECTOR of TOP-REAL n holds
    v in Lin(A) /\ Lin(A2) iff v in (0).TOP-REAL n
    proof
      let v be VECTOR of TOP-REAL n;
      hereby
        assume v in Lin(A) /\ Lin(A2);
        then
A14:    v in Lin(A) & v in Lin(A2) by RLSUB_2:3;
        then consider a be Real such that
A15:    v = a * p by RLVECT_4:8;
        v in Plane(p,0.TOP-REAL n) by A10,A14,RLVECT_5:20;
        then consider y be Point of TOP-REAL n such that
A16:    y = v & |( p,y-0.TOP-REAL n )| = 0;
        |( p,v )| = 0 by A2,A16,RLVECT_1:4;
        then
A17:    a * |( p,p )| = 0 by A15,EUCLID_2:20;
        |( p,p )| > 0 by A1,EUCLID_2:43;
        then a = 0 by A17;
        then v = 0.TOP-REAL n by A15,RLVECT_1:10;
        hence v in (0).TOP-REAL n by RLVECT_3:29;
      end;
      assume v in (0).TOP-REAL n;
      then v = 0.TOP-REAL n by RLVECT_3:29;
      hence v in Lin(A) /\ Lin(A2) by RLSUB_1:17;
    end;
    hence thesis by RLSUB_1:31;
  end;
  then dim TOP-REAL n = dim Lin A + dim Lin A2
    by RLVECT_5:37,A13,RLSUB_2:def 4;
  then n = dim Lin A + 1 by A11,RLAFFIN3:4;
  hence card A = n - 1 by RLVECT_5:29;
  thus [#]Lin(A) = Plane(p,0.TOP-REAL n) by A10,RLVECT_5:20;
end;
