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 Th27:
  i in Seg n implies Mx2Tran AxialSymmetry(i,n) is rotation
proof
  set S=Seg n,M=Mx2Tran AxialSymmetry(i,n);
  assume A1: i in S;
  let p be Point of TOP-REAL n;
  len p=n by CARD_1:def 7;
  then A2: i in dom p by A1,FINSEQ_1:def 3;
  thus|.M.p.|=sqrt Sum sqr(p+*(i,-p.i)) by A1,Th10
  .=sqrt((Sum sqr p)-(p.i)^2+(-p.i)^2) by A2,Th3
  .=|.p.|;
end;
