 reserve n,s for Nat;

theorem
  for s being 4_or_greater Nat,
      x being non zero s-gonal number holds
    Polygon (s, IndexPoly (s,x)) = x
  proof
    let s be 4_or_greater Nat,
        x be non zero s-gonal number;
A1: s - 2 <> 0 by EC_PF_2:def 1;
A2: 0 <= (((8 * s - 16) * x) + (s - 4) ^2) by Th35;
    set qq = sqrt (((8*s - 16) * x) + (s - 4) ^2);
    set w = IndexPoly (s,x);
A3: w ^2 * (s - 2) = ((qq + s - 4)^2 / (2 * s - 4) ^2) * (s - 2)
      by XCMPLX_1:76
     .= ((qq + s - 4)^2 / (4 * (s - 2) * (s - 2))) * (s - 2)
     .= ((qq^2 + (s - 4)^2 + 2*qq*(s-4))/ (4 * (s - 2))) by XCMPLX_1:92,A1;
A4: w * (s - 4) = (qq + s - 4) * (s - 4) / (2 * s - 4) by XCMPLX_1:74
      .= 2 * (qq * (s - 4) + (s - 4)^2) / (2 * (2*(s - 2))) by XCMPLX_1:91
      .= 2 * (qq * (s - 4) + (s - 4)^2) / (4 * (s - 2));
A5: qq^2 = ((8 * s - 16) * x) + (s - 4) ^2 by SQUARE_1:def 2,A2;
    Polygon (s, w) = ( (qq^2 + (s - 4)^2 + 2 * qq * (s - 4))
      - 2 * (qq * (s - 4) + (s - 4)^2)) / (4 * (s - 2)) / 2
        by XCMPLX_1:120,A3,A4
      .= ( (qq^2 + (s - 4)^2 + 2 * qq * (s - 4))
      - 2 * (qq * (s - 4) + (s - 4)^2)) / ((4 * (s - 2) * 2)) by XCMPLX_1:78
      .= x by A1,XCMPLX_1:89,A5;
    hence thesis;
  end;
