theorem Th31:
  not mt is one-to-one implies Det Segm(M,mt1,mt) = 0.K
proof
  assume not mt is one-to-one;
  then consider x,y being object such that
A1: x in dom mt and
A2: y in dom mt and
A3: mt.x=mt.y and
A4: x<>y;
A5: dom mt=Seg m by FINSEQ_2:124;
  then consider i be Nat such that
A6: x=i and
A7: 1<=i and
A8: i<=m by A1;
  consider j be Nat such that
A9: y=j and
A10: 1<=j and
A11: j<=m by A2,A5;
A12: j in Seg m by A10,A11;
  i in Seg m by A7,A8;
  hence thesis by A3,A4,A6,A9,A12,Th30;
end;
