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 Th10:
  i in Seg n implies (Mx2Tran AxialSymmetry(i,n)).p = p+*(i,-p.i)
proof
  set S=Seg n,Mp=(Mx2Tran AxialSymmetry(i,n)).p,p0=p+*(i,-p.i);
  A1: len p=n by CARD_1:def 7;
  assume A2: i in S;
  A3: for j st 1<=j & j<=n holds Mp.j=p0.j
  proof
    let j such that
    A4: 1<=j & j<=n;
    A5: j in S by A4;
    A6: j in dom p by A1,A4,FINSEQ_3:25;
    per cases;
    suppose A7: j<>i;
      then p0.j=p.j by FUNCT_7:32;
      hence thesis by A2,A7,Th8;
    end;
    suppose A8: j=i;
      then p0.j=-p.i by A6,FUNCT_7:31;
      hence thesis by A5,A8,Th9;
    end;
  end;
  len p0=len p & len Mp=n by CARD_1:def 7,FUNCT_7:97;
  hence thesis by A1,A3;
end;
