
theorem Th6:
for H being RealUnitarySpace holds
  ex F be Function of [:the carrier of H,(the carrier of H)* :],
          (the carrier of H)*
    st for x be Point of H, e be FinSequence of H
        ex Fe be FinSequence of H
         st Fe = F.(x,e) & Fe = (x.|.e) (*) e
proof
let H be RealUnitarySpace;
set CH = the carrier of H;
defpred P1[ object, object, object ] means
  ex x be Point of H,e be FinSequence of CH
    st $1=x & $2=e
       &
       ex Fe be FinSequence of CH
          st Fe = $3 & Fe = (x.|.e) (*) e;
A1: for x, y being object st x in CH & y in CH* holds
ex z being object st z in CH* & P1[x,y,z]
proof
  let x0, y0 be object;
  assume A2: x0 in CH & y0 in CH*; then
  reconsider xn=x0 as Point of H;
  reconsider e=y0 as FinSequence of CH by A2,FINSEQ_1:def 11;
  defpred P2[ object, object ] means
    $2= (xn.|. (e/.$1))*(e/.$1);
A3: for k being Nat st k in Seg len e holds
    ex z being Element of CH st P2[k,z];
  consider Fe being FinSequence of CH such that
 A4:dom Fe = Seg len e
 & for k being Nat st k in Seg len e holds
    P2[k,Fe . k] from FINSEQ_1:sch 5(A3);
A5: len Fe = len e by A4,FINSEQ_1:def 3;
A6: for i be Nat st 1 <= i & i <= len Fe holds
               Fe.i = (xn.|. (e/.i))*(e/.i)
proof
  let i be Nat;
  assume 1 <= i & i <= len Fe; then
  i in Seg len Fe;
  hence thesis by A4,A5;
end;
take Fe;
thus Fe in CH* by FINSEQ_1:def 11;
Fe = (xn.|.e)(*)(e)
proof
  thus len Fe = len ((xn.|.e)(*)(e)) by A5,DefR;
  let i be Nat such that
E1: 1 <= i & i <= len Fe;
  len (xn.|.e) = len e by DefSK;
  then i in dom (xn.|.e) by A5,E1,FINSEQ_3:25; then
E2: (xn.|.e)/.i = (xn.|.e).i by PARTFUN1:def 6;
  thus Fe.i = (xn.|. (e/.i))*(e/.i) by A6,E1
  .= (xn.|.e)/.i * (e/.i) by E1,E2,A5,DefSK
  .= ((xn.|.e)(*)e).i by E1,A5,DefR;
end;
hence thesis;
end;
consider F being Function of [:CH,CH*:],CH* such that
A7:for z, y being object st z in CH & y in CH*
    holds
       P1[z,y,F . (z,y)] from BINOP_1:sch 1(A1);
take F;
let z be Point of H,e0 be FinSequence of CH;
 e0 in CH* by FINSEQ_1:def 11; then
    P1[z,e0,F . (z,e0)] by A7;
hence thesis;
end;
