 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 Th22:
for G be RealNormSpace-Sequence,
    i be Element of dom G,
    x be Point of product G,
    r be Point of G.i holds
  reproj(i,x).r - x = reproj(i,0.(product G)).(r - proj(i).x) &
  x - reproj(i,x).r = reproj(i,0.(product G)).(proj(i).x - r)
proof
   let G be RealNormSpace-Sequence,
       i be Element of dom G,
       x be Point of product G,
       r be Point of G.i;
    set m=len G;
   reconsider xf=x as Element of product carr G by Th10;
A1:dom carr G = dom G by Lm1;
   reconsider Zr = 0.(product G) as Element of product carr G by Th10;
   reconsider ixr = reproj(i,x).r as Element of product carr G by Th10;
   reconsider p = reproj(i,x).r - x as Element of product carr G by Th10;
   reconsider q = reproj(i,0.(product G)).(r - proj(i).x)
                    as Element of product carr G by Th10;
A3:dom q = dom (carr G) by CARD_3:9;
   reconsider s = x - reproj(i,x).r as Element of product carr G by Th10;
   reconsider t = reproj(i,0.(product G)).(proj(i).x - r)
                   as Element of product carr G by Th10;
A5:dom t = dom carr G by CARD_3:9;
A6: reproj(i,x).r = x +* (i,r) by Def4;
   reconsider xfi =xf.i as Point of G.i;
A7:   reproj(i,0.(product G)).(r - proj(i).x)
              = 0.(product G) +* (i,(r - proj(i).x)) by Def4;
              then
A7a:
   q = Zr +* (i,(r- xfi)) by Def3;
A8:   reproj(i,0.(product G)).(proj(i).x -r)
            = 0.(product G) +* (i,(proj(i).x - r)) by Def4;
            then
A8a:
   t = Zr +* (i,(xfi -r)) by Def3;
   set ir= i .--> r;
   set irx1= (i .--> (r- xfi));
   set irx2= (i .--> (xfi - r));
    x in the carrier of product G; then
A9: x in product carr G by Th10;
    consider g1 be Function such that
A10:  x = g1 & dom g1 = dom carr G
    & for i be object st i in dom carr G holds g1.i in (carr G).i
        by A9,CARD_3:def 5;
   for k be object st k in dom p holds p.k = q.k
   proof
    let k be object;
    assume A11: k in dom p; then
    reconsider k0=k as Element of dom G by A1,CARD_3:9;
    consider g be Function such that
A12:  Zr = 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;
A13: k in dom Zr by A12,A11,CARD_3:9;
A14: k in dom x by A10,A11,CARD_3:9;
    per cases;
    suppose not k in {i}; then
A15:   k <> i by TARSKI:def 1; then
A16: q.k0 = Zr.k0 by A7,FUNCT_7:32;
     p.k = ixr.k0 -xf.k0 by Th15
        .= xf.k0 -xf.k0 by A15,A6,FUNCT_7:32; then
     p.k =0.(G.k0) by RLVECT_1:15;
     hence p.k = q.k by A16,Th14;
    end;
    suppose k in {i}; then
A17: k=i by TARSKI:def 1; then
A18: q.k0 = r- xfi by A7a,A13,FUNCT_7:31;
     p.k=ixr.k0 -xf.k0 by Th15;
     hence p.k = q.k by A18,A6,A17,A14,FUNCT_7:31;
    end;
   end;
   hence reproj(i,x).r - x = reproj(i,0.(product G)).(r - proj(i).x)
     by A3,FUNCT_1:2,CARD_3:9;
   for k be object st k in dom s holds s.k = t.k
   proof
    let k be object;
    assume A19: k in dom s; then
    reconsider k0=k as Element of dom G by A1,CARD_3:9;
    consider g be Function such that
A20:   Zr = 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;
A21: k in dom Zr by A20,A19,CARD_3:9;
A22: k in dom x by A10,A19,CARD_3:9;
    per cases;
    suppose not k in {i}; then
A23:   k <> i by TARSKI:def 1; then
A24: t.k0 = Zr.k0 by A8,FUNCT_7:32;
     s.k= xf.k0 - ixr.k0 by Th15
       .=xf.k0 -xf.k0 by A6,A23,FUNCT_7:32; then
     s.k =0.(G.k0) by RLVECT_1:15;
     hence s.k = t.k by A24,Th14;
    end;
    suppose k in {i}; then
A25: k=i by TARSKI:def 1; then
A26: t.k0 = xfi-r by A8a,A21,FUNCT_7:31;
     s.k=xf.k0 - ixr.k0 by Th15;
     hence s.k = t.k by A26,A6,A25,A22,FUNCT_7:31;
    end;
   end;
   hence thesis by A5,FUNCT_1:2,CARD_3:9;
end;
