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 Th8:
  i in Seg n & i <> j implies (Mx2Tran AxialSymmetry(i,n)).p.j = p.j
proof
  set A=AxialSymmetry(i,n),M=Mx2Tran A,Mp=M.p,S=Seg n;
  assume A1: i in S & i<>j;
  len Mp=n by CARD_1:def 7;
  then A2: dom Mp=S by FINSEQ_1:def 3;
  len p=n by CARD_1:def 7;
  then A3: dom p=S by FINSEQ_1:def 3;
  per cases;
    suppose A4: j in S;
      then 1<=j & j<=n by FINSEQ_1:1;
      hence Mp.j=@p"*"Col(A,j) by MATRTOP1:18
      .=p.j by A1,A4,Th5;
    end;
    suppose A5: not j in S;
      then Mp.j={} by A2,FUNCT_1:def 2;
      hence thesis by A3,A5,FUNCT_1:def 2;
    end;
end;
