reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th48:
  for k being positive Nat holds
  powersFS(-(k+1),r,2*(k+1)-1) =
  <* (-k) to_power r *> ^ powersFS(-k,r,2*k-1) ^ <* k to_power r *>
  proof
    let k be positive Nat;
    set F = powersFS(-(k+1),r,2*(k+1)-1);
    set f = powersFS(-k,r,2*k-1);
    set g = <*(-k) to_power r*>;
    set h = <*k to_power r*>;
A1: len F = 2*(k+1)-1 by Def7;
A2: len g = 1 by FINSEQ_1:39;
A3: len f = 2*k-1 by Def7;
A4: len h = 1 by FINSEQ_1:39;
A5: len(g^f) = len g + len f by FINSEQ_1:22;
    then len(g^f^h) = len g + len f + len h by FINSEQ_1:22;
    hence len F = len(g^f^h) by A2,A3,A4,Def7;
    let a such that
A6: 1 <= a & a <= len F;
A7: F.a = (-(k+1)+a) to_power r by A1,A6,Def7;
    per cases by A6,XXREAL_0:1;
    suppose a = 1;
      hence F.a = (g^(f^h)).a by A7
      .= (g^f^h).a by FINSEQ_1:32;
    end;
    suppose that
A8:   1 < a and
A9:  a < len F;
      a < len g + len f + 1 by A2,A3,A9,Def7;
      then
A10:  a <= len g + len f by NAT_1:13;
      then
A11:  a in dom(g^f) by A5,A8,FINSEQ_3:25;
A12:  len g + 1 <= a by A2,A8,NAT_1:13;
A13:  1 <= a-1 by A8,INT_1:52;
      thus F.a = (-k+(a-1)) to_power r by A7
      .= f.(a-1) by A2,A3,A8,A13,A10,XREAL_1:9,Def7
      .= (g^f).a by A2,A10,A12,FINSEQ_1:23
      .= (g^f^h).a by A11,FINSEQ_1:def 7;
    end;
    suppose
A14:  a = len F;
      then a = len(g^f)+1 by A2,A3,A5,Def7;
      hence (g^f^h).a = F.a by A1,A7,A14;
    end;
  end;
