 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;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem Th48:
for G be RealNormSpace-Sequence,
    i be Element of dom G, x,y be Point of product G,
    xi be Point of G.i
  st y = reproj(i,x).xi holds reproj(i,x)=reproj(i,y)
proof
   let G be RealNormSpace-Sequence,
       i be Element of dom G, x,y be Point of product G,
       xi be Point of G.i;
   assume A1: y = reproj(i,x).xi;
   for v be Element of G.i holds reproj(i,x).v = reproj(i,y).v
   proof
    let v be Element of G.i;
A2: reproj(i,x).v = x +* (i,v) & reproj(i,y).v = y +* (i,v) by Def4;
    reconsider xv = reproj(i,x).v, yv = reproj(i,y).v
       as (len G)-element FinSequence;
A3: dom xv = Seg len G & dom yv = Seg len G by FINSEQ_1:89; then
A4: dom xv = dom G by FINSEQ_1:def 3;

    for k be Nat st k in dom xv holds xv.k = yv.k
    proof
     let k be Nat;
     assume A5: k in dom xv;
     x in the carrier of product G & y in the carrier of product G; then
A6:  x in product carr G & y in product carr G by Th10; then
     consider g be Function such that
A7:   x = 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;
     consider g1 be Function such that
A8:   y = g1 & dom g1 = dom carr G
      & for i be object st i in dom carr G holds g1.i in (carr G).i
        by A6,CARD_3:def 5;
A9:  k in dom y & k in dom x by A7,A8,Lm1,A5,A4;
     per cases;
     suppose k = i; then
      (y +*(i,v)).k = v & (x +*(i,v)).k = v by A9,FUNCT_7:31;
      hence yv.k = xv.k by A2;
     end;
     suppose A10: k <> i;
A11:   yv.k = y.k & xv.k = x.k by A2,A10,FUNCT_7:32;
      y = x +* (i,xi) by A1,Def4;
      hence yv.k = xv.k by A11,A10,FUNCT_7:32;
      end;
     end;
     hence reproj(i,x).v = reproj(i,y).v by A3,FINSEQ_1:13;
    end;
    hence thesis;
end;
