reserve x, y, X for set;
reserve E for non empty set;
reserve e for Element of E;
reserve u, u1, v, v1, v2, w, w9, w1, w2 for Element of E^omega;
reserve F for Subset of E^omega;
reserve i, k, l for Nat;
reserve TS for non empty transition-system over F;
reserve S, T for Subset of TS;

theorem Th5:
  k <> 0 & len u <= k + 1 implies ex v1, v2 st len v1 <= k & len v2
  <= k & u = v1^v2
proof
  assume that
A1: k <> 0 and
A2: len u <= k + 1;
  per cases;
  suppose
    len u = k + 1;
    then consider v1, e such that
A3: len v1 = k and
A4: u = v1 ^ <%e%> by FLANG_1:7;
    reconsider v2 = <%e%> as Element of E^omega;
    take v1, v2;
    thus len v1 <= k by A3;
    0 + 1 <= k by A1,NAT_1:13;
    hence len v2 <= k by AFINSQ_1:34;
    thus u = v1^v2 by A4;
  end;
  suppose
A5: len u <> k + 1;
    reconsider v2 = <%>E as Element of E^omega;
    take u, v2;
    thus len u <= k by A2,A5,NAT_1:8;
    thus len v2 <= k;
    thus u = u^{}
      .= u^v2;
  end;
end;
