let s be State of ; :: thesis: ( s starts_at 0 & Euclide-Algorithm c= s implies for x, y being Integer st s . (dl. 0 ) = x & s . (dl. 1) = y & x > 0 & y > 0 holds
(Result s) . (dl. 0 ) = x gcd y )
assume that
A1: s starts_at 0 and
A2: Euclide-Algorithm c= s ; :: thesis: for x, y being Integer st s . (dl. 0 ) = x & s . (dl. 1) = y & x > 0 & y > 0 holds
(Result s) . (dl. 0 ) = x gcd y
let x, y be Integer; :: thesis: ( s . (dl. 0 ) = x & s . (dl. 1) = y & x > 0 & y > 0 implies (Result s) . (dl. 0 ) = x gcd y )
assume that
A3: ( s . (dl. 0 ) = x & s . (dl. 1) = y ) and
A4: x > 0 and
A5: y > 0 ; :: thesis: (Result s) . (dl. 0 ) = x gcd y
A6: abs y = y by A5, ABSVALUE:def 1;

now

per cases ( x > y or x = y or y > x ) by XXREAL_0:1;

case x > y ; :: thesis: (Result s) . (dl. 0 ) = x gcd y

hence (Result s) . (dl. 0 ) = x gcd y by A1, A2, A3, A5, Lm6; :: thesis: verum

end;

case A7: x = y ; :: thesis: ex s' being Element of K243(the Object-Kind of SCM ) st (Result s) . (dl. 0 ) = x gcd y

reconsider x' = x, y' = y as Element of NAT by A4, A5, INT_1:16;
take s' = Computation s,4; :: thesis: (Result s) . (dl. 0 ) = x gcd y
A8: s = Computation s,(4 * 0 ) by AMI_1:13;
A9: s' = Computation s,(4 * (0 + 1)) ;
x mod y = x' mod y'
.= 0 by A7, NAT_D:25 ;
then s' . (dl. 1) = 0 by A1, A2, A3, A4, A5, A8, A9, Lm5;
then IC s' = 4 by A1, A2, A3, A4, A5, A9, Lm4;
then s halts_at IC s' by A2, Lm3;
hence (Result s) . (dl. 0 ) = s' . (dl. 0 ) by AMI_1:87
.= y by A1, A2, A3, A4, A5, A8, A9, Lm5
.= x gcd y by A6, A7, NAT_D:32 ;
:: thesis: verum

end;

case A10: y > x ; :: thesis: ex s' being Element of K243(the Object-Kind of SCM ) st (Result s) . (dl. 0 ) = x gcd y

reconsider x' = x, y' = y as Element of NAT by A4, A5, INT_1:16;
take s' = Computation s,4; :: thesis: (Result s) . (dl. 0 ) = x gcd y
A11: s' = Computation s,(4 * (0 + 1)) ;
A12: s = Computation s,(4 * 0 ) by AMI_1:13;
then A13: s' . (dl. 0 ) = y by A1, A2, A3, A4, A5, A11, Lm5;
A14: Euclide-Algorithm c= s' by A2, AMI_1:86;
x mod y = x' mod y'
.= x' by A10, NAT_D:24 ;
then A15: s' . (dl. 1) = x by A1, A2, A3, A4, A5, A12, A11, Lm5;
then IC s' = 0 by A1, A2, A3, A4, A5, A11, Lm4;
then A16: s' starts_at 0 by AMI_1:def 41;
then consider k0 being Element of NAT such that
A17: s' halts_at IC (Computation s',k0) by A4, A10, A13, A15, A14, Lm6;
Computation s,(k0 + 4) = Computation s',k0 by AMI_1:51;
then A18: s halts_at IC (Computation s,(k0 + 4)) by A17, AMI_1:91;
(Result s') . (dl. 0 ) = x gcd y by A4, A10, A13, A15, A16, A14, Lm6;
hence (Result s) . (dl. 0 ) = x gcd y by A18, AMI_1:90; :: thesis: verum

end;

end;

end;

hence (Result s) . (dl. 0 ) = x gcd y ; :: thesis: verum