let R be good Ring; :: thesis: for k being natural number holds il. (SCM R),k = il. k
let k be natural number ; :: thesis: il. (SCM R),k = il. k
deffunc H₁( Element of NAT ) -> Instruction-Location of SCM = il. $1;
A1: for x being Element of NAT holds H₁(x) is Element of NAT ;
consider f being Function of NAT , NAT such that
A2: for k being Element of NAT holds f . k = H₁(k) from FUNCT_2:sch 9(A1);
reconsider f = f as IL-Function of NAT , SCM R by AMI_1:def 36;
A3: f is bijective by A2, Th65;
A4: for k being Element of NAT holds
( f . (k + 1) in SUCC (f . k) & ( for j being Element of NAT st f . j in SUCC (f . k) holds
k <= j ) ) by A2, Th65;
reconsider k = k as Element of NAT by ORDINAL1:def 13;
ex f being IL-Function of NAT , SCM R st
( f is bijective & ( for m, n being Element of NAT holds
( m <= n iff f . m <= f . n ) ) & il. k = f . k )

proof

take f ; :: thesis: ( f is bijective & ( for m, n being Element of NAT holds
( m <= n iff f . m <= f . n ) ) & il. k = f . k )
thus f is bijective by A2, Th65; :: thesis: ( ( for m, n being Element of NAT holds
( m <= n iff f . m <= f . n ) ) & il. k = f . k )
thus for m, n being Element of NAT holds
( m <= n iff f . m <= f . n ) by A3, A4, AMISTD_1:18; :: thesis: il. k = f . k
thus il. k = f . k by A2; :: thesis: verum

end;

hence il. (SCM R),k = il. k by AMISTD_1:def 12; :: thesis: verum