theorem Th54:
  not F in Permutations(n) implies Det(A*F) = 0.K
proof
  assume not F in Permutations(n);
  then
A1: F is not onto or F is not one-to-one by MATRIX_1:def 12;
  card Seg n=card Seg n;
  then F is not one-to-one by A1,FINSEQ_4:63;
  then consider x,y being object such that
A2: x in dom F and
A3: y in dom F and
A4: F.x = F.y and
A5: x<>y by FUNCT_1:def 4;
A6: dom F=Seg n by FUNCT_2:52;
  then reconsider x,y as Nat by A2,A3;
  Line(A*F,x)=A.(F.x) by A2,A6,Th38;
  then
A7: Line(A*F,x)=Line(A*F,y) by A3,A4,A6,Th38;
  x>y or y>x by A5,XXREAL_0:1;
  hence thesis by A2,A3,A6,A7,Th50;
end;
