 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;

theorem Th12:
for G be RealNormSpace-Sequence, p,q be Point of product G,
    r0,p0,q0 be Element of product carr G st p=p0 & q=q0
holds p+q = r0
  iff for i be Element of dom G holds r0.i = p0.i + q0.i
proof
   let G be RealNormSpace-Sequence, p,q be Point of product G,
       r0,p0,q0 be Element of product carr G;
   assume A1: p=p0 & q=q0;
   len carr G = len G by PRVECT_1:def 11; then
A2:dom carr G = Seg len G by FINSEQ_1:def 3
             .= dom G by FINSEQ_1:def 3;
A3: product G = NORMSTR(# product carr G,zeros G,[:addop G:],[:multop G:],
    productnorm G #) by PRVECT_2:6;
   hereby assume A4: p+q = r0;
    hereby let i be Element of dom G;
     reconsider i0=i as Element of dom carr G by A2;
     (addop G).i0 = the addF of (G.i0) by PRVECT_1:def 12;
     hence r0.i = p0.i + q0.i by A1,A4,A3,PRVECT_1:def 8;
    end;
   end;
   assume A5: for i be Element of dom G holds r0.i = p0.i + q0.i;
   reconsider pq=p+q as Element of product carr G by Th10;
A6:ex g be Function st
     pq = g & dom g = dom carr G
   & for i be object st i in dom carr G holds g.i in (carr G).i
         by CARD_3:def 5;
A7:ex g be Function st
     r0 = g & dom g = dom carr G
   & for i be object st i in dom carr G holds g.i in (carr G).i
         by CARD_3:def 5;
   now let i0 be object;
    assume A8: i0 in dom pq; then
    reconsider i1=i0 as Element of dom G by A2,A6;
    reconsider i =i0 as Element of dom carr G by A8,A6;
    (addop G).i = the addF of (G.i) by PRVECT_1:def 12; then
    pq.i0 = p0.i1 + q0.i1 by A1,A3,PRVECT_1:def 8;
    hence pq.i0 = r0.i0 by A5;
   end;
   hence p+q = r0 by A6,A7,FUNCT_1:2;
end;
