reserve x,X for set,
        r,r1,r2,s for Real,
        i,j,k,m,n for Nat;
reserve p,q for Point of TOP-REAL n;

theorem Th6:
  i in Seg n implies @p"*"Col(AxialSymmetry(i,n),i) = -p.i
proof
  set S=Seg n;
  assume A1: i in S;
  reconsider pI=@p.i as Element of F_Real by XREAL_0:def 1;
  set A=AxialSymmetry(i,n),C=Col(A,i);
  A2: len A=n by MATRIX_0:25;
  then A3: dom A=S by FINSEQ_1:def 3;
  then A4: C.i=A*(i,i) by A1,MATRIX_0:def 8;
  len p=n & len C=n by A2,CARD_1:def 7;
  then len mlt(@p,C)=n by MATRIX_3:6;
  then A5: dom mlt(@p,C)=S by FINSEQ_1:def 3;
  A6: Indices A=[:S,S:] by MATRIX_0:24;
  A7: for k st k in dom mlt(@p,C) & k<>i holds mlt(@p,C).k=0.F_Real
  proof
    let k;
    assume that
    A8: k in dom mlt(@p,C) and
    A9: k<>i;
    @p.k in REAL by XREAL_0:def 1;
    then reconsider pk=@p.k as Element of F_Real;
    A10: [k,i] in Indices A by A1,A5,A6,A8,ZFMISC_1:87;
    C.k=A*(k,i) by A3,A5,A8,MATRIX_0:def 8;
    hence mlt(@p,C).k=pk*(A*(k,i)) by A8,FVSUM_1:60
    .=pk*0.F_Real by A1,A9,A10,Def2
    .=0.F_Real;
  end;
  thus@p"*"C=mlt(@p,C).i by A1,A5,A7,MATRIX_3:12
  .=pI*(A*(i,i)) by A1,A4,A5,FVSUM_1:60
  .=pI*(-1.F_Real) by A1,Def2
  .= pI *(-1)
  .=-p.i;
end;
