reconsider I = [(ProgramPart P)] as Program of ;
let k1, k2 be Element of NAT ; :: thesis: ( IC (Computation (s +* (P +* (Start-At (insloc 0 )))),k1) = insloc (card (ProgramPart P)) & ( for n being Element of NAT st not IC (Computation (s +* (P +* (Start-At (insloc 0 )))),n) in dom P holds
k1 <= n ) & IC (Computation (s +* (P +* (Start-At (insloc 0 )))),k2) = insloc (card (ProgramPart P)) & ( for n being Element of NAT st not IC (Computation (s +* (P +* (Start-At (insloc 0 )))),n) in dom P holds
k2 <= n ) implies k1 = k2 )
assume that
A3: IC (Computation (s +* (P +* (Start-At (insloc 0 )))),k1) = insloc (card (ProgramPart P)) and
A4: ( ( for n being Element of NAT st not IC (Computation (s +* (P +* (Start-At (insloc 0 )))),n) in dom P holds
k1 <= n ) & IC (Computation (s +* (P +* (Start-At (insloc 0 )))),k2) = insloc (card (ProgramPart P)) ) and
A5: for n being Element of NAT st not IC (Computation (s +* (P +* (Start-At (insloc 0 )))),n) in dom P holds
k2 <= n ; :: thesis: k1 = k2

A6: now

assume k2 < k1 ; :: thesis: contradiction
then insloc (card I) in dom I by A4, AMI_1:106;
hence contradiction by SCMFSA6A:15; :: thesis: verum

end;

now

assume k1 < k2 ; :: thesis: contradiction
then insloc (card I) in dom I by A3, A5, AMI_1:106;
hence contradiction by SCMFSA6A:15; :: thesis: verum

end;

hence k1 = k2 by A6, XXREAL_0:1; :: thesis: verum