reserve x,y for object,X,Y for set,
  D for non empty set,
  i,j,k,l,m,n,m9,n9 for Nat,
  i0,j0,n0,m0 for non zero Nat,
  K for Field,
  a,b for Element of K,
  p for FinSequence of K,
  M for Matrix of n,K;
reserve A for (Matrix of D),
  A9 for Matrix of n9,m9,D,
  M9 for Matrix of n9, m9,K,
  nt,nt1,nt2 for Element of n-tuples_on NAT,
  mt,mt1 for Element of m -tuples_on NAT,
  M for Matrix of K;
reserve P,P1,P2,Q,Q1,Q2 for without_zero finite Subset of NAT;
reserve v,v1,v2,u,w for Vector of n-VectSp_over K,
  t,t1,t2 for Element of n -tuples_on the carrier of K,

  L for Linear_Combination of n-VectSp_over K,
  M,M1 for Matrix of m,n,K;

theorem Th120:
  P c= Seg m & M is without_repeated_line implies Segm(M,P,Seg n)
  is without_repeated_line
proof
  assume that
A1: P c= Seg m and
A2: M is without_repeated_line;
  set S=Segm(M,P,Seg n);
  let x1,x2 be object such that
A3: x1 in dom S and
A4: x2 in dom S and
A5: S.x1=S.x2;
  reconsider i1=x1,i2=x2 as Element of NAT by A3,A4;
  len S=card P by MATRIX_0:def 2;
  then
A6: dom S=Seg card P by FINSEQ_1:def 3;
  then
A7: Line(S,i1)=S.i1 by A3,MATRIX_0:52;
A8: Line(S,i2)=S.i2 by A4,A6,MATRIX_0:52;
A9: Sgm P is one-to-one by FINSEQ_3:92;
A10: dom Sgm P=dom S by A6,FINSEQ_3:40;
  Seg m<>{} by A1,A3,A6;
  then m <> 0;
  then
A11: width M=n by Th1;
  then
A12: Line(S,i1) = Line(M,Sgm P.i1) by A3,A6,Th48;
A13: Line(S,i2) = Line(M,Sgm P.i2) by A4,A6,A11,Th48;
A14: len M=m by MATRIX_0:def 2;
A15: rng Sgm P=P by FINSEQ_1:def 14;
  then
A16: Sgm P.i2 in P by A4,A10,FUNCT_1:def 3;
  then
A17: Line(M,Sgm P.i2)=M.(Sgm P.i2) by A1,MATRIX_0:52;
A18: Sgm P.i1 in P by A3,A10,A15,FUNCT_1:def 3;
  then Sgm P.i1 in Seg m by A1;
  then
A19: Sgm P.i1 in dom M by A14,FINSEQ_1:def 3;
  Sgm P.i2 in Seg m by A1,A16;
  then
A20: Sgm P.i2 in dom M by A14,FINSEQ_1:def 3;
  Line(M,Sgm P.i1)=M.(Sgm P.i1) by A1,A18,MATRIX_0:52;
  then Sgm P.i1=Sgm P.i2 by A2,A5,A12,A13,A7,A8,A17,A19,A20;
  hence thesis by A3,A4,A10,A9;
end;
