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 Th9:
  i in Seg n implies (Mx2Tran AxialSymmetry(i,n)).p.i = -p.i
proof
  set A=AxialSymmetry(i,n),M=Mx2Tran A,Mp=M.p,S=Seg n;
  assume A1: i in S;
  then 1<=i & i<=n by FINSEQ_1:1;
  hence Mp.i=@p"*"Col(A,i) by MATRTOP1:18
  .=-p.i by A1,Th6;
end;
