reserve i, j, m, n, k for Nat,
  x, y for set,
  K for Field,
  a,a1 for Element of K;
reserve V1,V2,V3 for finite-dimensional VectSp of K,
  f for Function of V1,V2,

  b1,b19 for OrdBasis of V1,
  B1 for FinSequence of V1,
  b2 for OrdBasis of V2,
  B2 for FinSequence of V2,

  B3 for FinSequence of V3,
  v1,w1 for Element of V1,
  R,R1,R2 for FinSequence of V1,
  p,p1,p2 for FinSequence of K;

theorem
  for A,B be Matrix of len b1,len B2,K holds 
  Mx2Tran(A+B,b1,B2) = Mx2Tran(A,b1,B2) + Mx2Tran(B,b1,B2)
proof
  let A,B be Matrix of len b1,len B2,K;
  set AB=A+B;
  set M=Mx2Tran(A+B,b1,B2);
  set MA=Mx2Tran(A,b1,B2);
  set MB=Mx2Tran(B,b1,B2);
  now
    let x be object such that
A1: x in the carrier of V1;
    reconsider v=x as Element of V1 by A1;
    now
      per cases;
      suppose
A2:     len b1=0;
        hence M.x = 0.V2 by A1,Th33
          .= 0.V2+0.V2 by RLVECT_1:def 4
          .= MA.v+0.V2 by A2,Th33
          .= MA.v+MB.v by A2,Th33
          .= (MA+MB).x by MATRLIN:def 3;
      end;
      suppose
A3:     len b1>0;
        set L=LineVec2Mx(v|--b1);
A4:     width L=len (v|--b1) & len (v|--b1)=len b1 by MATRIX_0:23,MATRLIN:def 7
;
        set mB=lmlt(Line(L*B,1),B2);
A5:     len B=len b1 & width B=len B2 by A3,MATRIX_0:23;
        then
A6:     width (L*B)=len B2 by A4,MATRIX_3:def 4;
        then len Line(L*B,1)=len B2 by CARD_1:def 7;
        then dom Line(L*B,1)=dom B2 by FINSEQ_3:29;
        then
A7:     dom mB=dom B2 by MATRLIN:12;
        then
A8:     len mB=len B2 by FINSEQ_3:29;
A9:     len A=len b1 by A3,MATRIX_0:23;
        len L=1 by MATRIX_0:23;
        then
A11:    len (L*A)=1 by A9,A4,MATRIX_3:def 4;
        set mA=lmlt(Line(L*A,1),B2);
A12:    width A=len B2 by A3,MATRIX_0:23;
        then
A13:    width (L*A)=len B2 by A9,A4,MATRIX_3:def 4;
        then len Line(L*A,1)=len B2 by CARD_1:def 7;
        then dom Line(L*A,1)=dom B2 by FINSEQ_3:29;
        then
A14:    dom mA=dom B2 by MATRLIN:12;
        then
A15:    len mA=len B2 by FINSEQ_3:29;
A16:    dom (mA+mB) = dom B2/\dom B2 by A14,A7,Lm3
          .= dom B2;
        then
A17:    len (mA+mB)=len B2 by FINSEQ_3:29;
A18:    now
          let k be Nat such that
A19:      k in dom mA;
          mA/.k=mA.k & mB/.k=mB.k by A14,A7,A19,PARTFUN1:def 6;
          hence (mA+mB).k = mA/.k + mB/.k by A14,A16,A19,FVSUM_1:17;
        end;
        thus M.x = Sum lmlt (Line(L * AB,1),B2) by Def3
          .= Sum lmlt (Line(L * A+L*B,1),B2) by A9,A12,A5,A4,MATRIX_4:62
          .= Sum lmlt (Line(L * A,1)+Line(L*B,1),B2)by A11,A13,A6,Lm5
          .= Sum (mA + mB) by Th7
          .= Sum mA +Sum mB by A15,A8,A17,A18,RLVECT_2:2
          .= MA.v+Sum mB by Def3
          .= MA.v+MB.v by Def3
          .= (MA+MB).x by MATRLIN:def 3;
      end;
    end;
    hence M.x=(MA+MB).x;
  end;
  hence thesis by FUNCT_2:12;
end;
