reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th25:
for m be non zero Nat, r be Real,
   w,u be FinSequence of REAL m st
  u = r(#)w holds Sum u = r * Sum w
proof
   let m be non zero Nat, r be Real;
   defpred P[Nat] means
    for xseq, yseq be FinSequence of REAL m st
     $1= len xseq & len xseq = len yseq &
     for i be Nat st i in dom xseq holds yseq/.i=r*(xseq/.i)
        holds Sum yseq = r * Sum xseq;
A1:P[0]
   proof
    let xseq,yseq be FinSequence of REAL m;
    assume A2: 0 = len xseq & len xseq = len yseq & for i be Nat st
                 i in dom xseq holds yseq/.i=r*(xseq/.i);
    reconsider r1=r as Real;
     Sum xseq = 0*m by A2,EUCLID_7:def 11;
    then r * Sum xseq = r1 * 0.TOP-REAL m by EUCLID:70
                .= 0.TOP-REAL m by RLVECT_1:10
                .= 0*m by EUCLID:70;
    hence thesis by A2,EUCLID_7:def 11;
   end;
A3:now let i be Nat;
    assume A4: P[i];
     now let xseq,yseq be FinSequence of REAL m;
     assume A5: i+1 = len xseq & len xseq = len yseq &
        for k be Nat st
          k in dom xseq holds yseq/.k = r * xseq/.k;
then A6:dom xseq = dom yseq by FINSEQ_3:29;
     set xseq0 = xseq|i;
     set yseq0 = yseq|i;
A7: i = len xseq0 by A5,FINSEQ_1:59,NAT_1:11;
then A8: len xseq0 = len yseq0 by A5,FINSEQ_1:59,NAT_1:11;
      for k be Nat st k in dom xseq0 holds yseq0/.k=r*(xseq0/.k)
     proof
      let k be Nat;
      assume A9: k in dom xseq0;
then A10:  k in dom xseq & k in Seg i by RELAT_1:57;
A11: k in dom(yseq|Seg i) by A9,A8,FINSEQ_3:29;
A12:  xseq/.k = xseq.k by A10,PARTFUN1:def 6
             .= (xseq|Seg i).k by A10,FUNCT_1:49
             .= xseq0/.k by A9,PARTFUN1:def 6;
       yseq0/.k = (yseq|Seg i).k by A11,PARTFUN1:def 6
              .= yseq.k by A10,FUNCT_1:49
              .= yseq/.k by A10,A6,PARTFUN1:def 6;
      hence yseq0/.k = r * (xseq0/.k) by A5,A10,A12;
     end;
then A13:Sum yseq0 = r * Sum xseq0 by A7,A8,A4;
     consider v be Element of REAL m such that
A14: v=xseq.(len xseq) & Sum xseq = Sum xseq0 + v by A5,A7,PDIFF_6:15;
     consider w be Element of REAL m such that
A15:w=yseq.(len yseq) & Sum yseq = Sum yseq0 + w by A5,A7,A8,PDIFF_6:15;
A16:dom xseq = Seg(i+1) by A5,FINSEQ_1:def 3;
then A17:len yseq in dom xseq by A5,FINSEQ_1:4;
then w = yseq/.(len yseq) by A15,A6,PARTFUN1:def 6
      .= r * xseq/.(len yseq) by A5,A16,FINSEQ_1:4
      .= r * v by A17,A5,A14,PARTFUN1:def 6;
     hence Sum yseq = r * Sum xseq by A15,A13,A14,RVSUM_1:51;
    end;
    hence P[i+1];
   end;
A18:for k be Nat holds P[k] from NAT_1:sch 2(A1,A3);
   let xseq,yseq be FinSequence of REAL m;
A19: r(#)xseq = xseq[#]r by INTEGR15:def 11;
   assume A20: yseq=r(#)xseq;
then A21:dom yseq = dom xseq by A19,VALUED_2:def 39;
then A22:len xseq = len yseq by FINSEQ_3:29;
    for i be Nat st i in dom xseq holds yseq/.i = r * xseq/.i
      by A20,A21,INTEGR15:23;
   hence Sum yseq = r * Sum xseq by A22,A18;
end;
