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 Th4:
  i in Seg n implies Det AxialSymmetry(i,n) = -1.F_Real
proof
  assume A1: i in Seg n;
  then consider M be Matrix of n,F_Real such that
  A2: Det M=-1.F_Real and
  A3: M*(i,i)=-1.F_Real & for k,m st[k,m] in Indices M holds
    (k=m & k<>i implies M*(k,k)=1.F_Real) & (k<>m implies M*(k,m)=0.F_Real)
    by Lm1;
  Det M<>0.F_Real by A2;
  then M is invertible by LAPLACE:34;
  hence thesis by A1,A2,A3,Def2;
end;
