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;

theorem Th54:
  X c= P implies Sgm X = Sgm P * Sgm (Sgm P " X)
proof
  assume
A1: X c= P;
A2: Sgm P " X c= dom Sgm P by RELAT_1:132;
A3:  P is included_in_Seg;
A4: rng Sgm P=P by FINSEQ_1:def 14;
  rng ((Sgm P) | (Sgm P " X)) = (Sgm P).:(Sgm P " X) by RELAT_1:115
    .= X by A1,A4,FUNCT_1:77;
  hence thesis by A3,A2,FINSEQ_6:129;
end;
