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 Th5:
  i in Seg n & j in Seg n & i<>j implies @p"*"Col(AxialSymmetry(i,n),j)=p.j
proof
  set S=Seg n;
  assume that
  A1: i in S and
  A2: j in S and
  A3: i<>j;
  set A=AxialSymmetry(i,n),C=Col(A,j);
  A4: Indices A=[:S,S:] by MATRIX_0:24;
  then A5: [j,j] in Indices A by A2,ZFMISC_1:87;
  A6: len A=n by MATRIX_0:25;
  then A7: dom A=S by FINSEQ_1:def 3;
  len C=n by A6,MATRIX_0:def 8;
  then A8: dom C=S by FINSEQ_1:def 3;
  A9: now let m such that
     A10: m in dom C and
     A11: m<>j;
     A12: [m,j] in Indices A by A2,A4,A8,A10,ZFMISC_1:87;
     thus C.m=A*(m,j) by A7,A8,A10,MATRIX_0:def 8
     .=0.F_Real by A1,A11,A12,Def2;
  end;
  len p=n by CARD_1:def 7;
  then A13: dom p=S by FINSEQ_1:def 3;
  C.j=A*(j,j) by A2,A7,MATRIX_0:def 8
  .=1.F_Real by A1,A3,A5,Def2;
  hence p.j=Sum(mlt(C,@p)) by A2,A8,A9,A13,MATRIX_3:17
  .=@p"*"C by FVSUM_1:64;
end;
