reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);
reserve D for non empty set,
  bD for FinSequence of D,
  b,f,g for FinSequence of K,
  MD for Matrix of D;

theorem Th72:
  for A,B be Matrix of K st width A = len B & ( width A = 0
  implies len A = 0 ) & ( width B = 0 implies len B = 0 ) holds
  Space_of_Solutions_of B is Subspace of Space_of_Solutions_of (A*B)
proof
  let A,B be Matrix of K such that
A1: width A = len B and
A2: width A = 0 implies len A = 0 and
A3: width B = 0 implies len B = 0;
  set AB=A*B;
A4: len AB=len A by A1,MATRIX_3:def 4;
A5: width AB=width B by A1,MATRIX_3:def 4;
  then reconsider AB as Matrix of len A,width B,K by A4,MATRIX_0:51;
  the carrier of Space_of_Solutions_of B c= the carrier of
  Space_of_Solutions_of AB
  proof
    let x be object;
    assume x in the carrier of Space_of_Solutions_of B;
    then x in Solutions_of(B,len B|->0.K) by A3,Def5;
    then consider f such that
A6: f=x and
A7: ColVec2Mx f in Solutions_of(B,ColVec2Mx (len B|->0.K));
    consider X such that
A8: X=ColVec2Mx f and
A9: len X = width B and
A10: width X = width ColVec2Mx (len B|->0.K) and
A11: B * X = ColVec2Mx (len B|->0.K) by A7;
A12: ColVec2Mx (len AB|->0.K)=0.(K,len A,1) by A4,Th32;
A13: ColVec2Mx (len B|->0.K)=0.(K,len B,1) by Th32;
    now
      per cases;
      suppose
        len A=0;
        then
        Solutions_of(AB,ColVec2Mx (len AB|->0.K)) ={{}} & X ={} by A4,A5,A9,A12
,Th51,MATRIX_0:def 3;
        hence X in Solutions_of(AB,ColVec2Mx (len AB|->0.K)) by TARSKI:def 1;
      end;
      suppose
A14:    len A<>0;
        then
A15:    width ColVec2Mx (len AB|->0.K)=1 by A4,Th26
          .= width ColVec2Mx (len B|->0.K) by A1,A2,A14,Th26;
        ColVec2Mx (len AB|->0.K) = A*(B*X) by A1,A2,A11,A13,A12,A14,MATRIXR2:18
          .= AB*X by A1,A9,MATRIX_3:33;
        hence X in Solutions_of(AB,ColVec2Mx (len AB|->0.K)) by A5,A9,A10,A15;
      end;
    end;
    then f in Solutions_of(AB,len AB|->0.K) by A8;
    hence thesis by A1,A2,A3,A4,A5,A6,Def5;
  end;
  hence thesis by A5,VECTSP_4:27;
end;
