reserve k,n,i,j for Nat;

theorem Th28:
  for n being Nat,K being commutative Ring, p being Element of
  Permutations(n), x being Element of K st n>=1 holds -(x,p) = -(x,p")
proof
  let n be Nat,K be commutative Ring, p be Element of Permutations(n), x be
  Element of K;
  assume
A1: n>=1;
  reconsider pf=p as Permutation of Seg n by MATRIX_1:def 12;
A2: len ((Permutations(n)))=n by MATRIX_1:9;
  per cases;
  suppose
    p is even;
    then -(x,p) = x & pf" is even by A1,A2,Th27,MATRIX_1:def 16;
    hence thesis by A2,MATRIX_1:def 16;
  end;
  suppose
    not p is even;
    then -(x,p) = -x & not p" is even by A1,Th27,MATRIX_1:def 16;
    hence thesis by MATRIX_1:def 16;
  end;
end;
