reserve m,n for Nat,
  a,b for Int_position,
  i,j for Instruction of SCMPDS,
  s,s1,s2 for State of SCMPDS,
  I,J for Program of SCMPDS;

theorem Th10:
  GCD-Algorithm. 0=GBP:=0 & GCD-Algorithm. 1=SBP:= 7 &
  GCD-Algorithm. 2=saveIC(SBP,RetIC) & GCD-Algorithm. 3=goto 2 &
  GCD-Algorithm. 4=halt SCMPDS &
  GCD-Algorithm. 5=(SBP,3)<=0_goto 9 &
  GCD-Algorithm. 6=(SBP,6):=(SBP,3) &
  GCD-Algorithm. 7=Divide(SBP,2,SBP,3) &
  GCD-Algorithm. 8=(SBP,7):=(SBP,3) &
  GCD-Algorithm. 9=(SBP,4+RetSP):=(GBP,1) &
  GCD-Algorithm. 10=AddTo(GBP,1,4) &
  GCD-Algorithm. 11=saveIC(SBP,RetIC) &
  GCD-Algorithm. 12=goto -7 & GCD-Algorithm. 13=(SBP,2):=(SBP,6) &
  GCD-Algorithm. 14=return SBP
proof
  set I2=i00 ';' i01, I3=I2 ';' i02, I4=I3 ';' i03, I5=I4 ';' i04,
  I6=I5 ';' i05, I7=I6 ';' i06, I8=I7 ';' i07, I9=I8 ';' i08, I10=I9 ';' i09,
  I11=I10 ';' i10, I12=I11 ';' i11, I13=I12 ';' i12, I14=I13 ';' i13;
A1: card I2=2 by Th5;
  then
A2: card I3=2+1 by Th4;
  then
A3: card I4=3+1 by Th4;
  then
A4: card I5=4+1 by Th4;
  then
A5: card I6=5+1 by Th4;
  then
A6: card I7=6+1 by Th4;
  then
A7: card I8=7+1 by Th4;
  then
A8: card I9=8+1 by Th4;
  then
A9: card I10=9+1 by Th4;
  then
A10: card I11=10+1 by Th4;
  then
A11: card I12=11+1 by Th4;
  then
A12: card I13=12+1 by Th4;
  then
A13: card I14=13+1 by Th4;
  set J14=i13 ';' i14, J13=i12 ';' J14, J12=i11 ';' J13, J11=i10 ';' J12,
  J10=i09 ';' J11, J9=i08 ';' J10, J8=i07 ';' J9, J7=i06 ';' J8,
  J6=i05 ';' J7, J5=i04 ';' J6, J4=i03 ';' J5, J3=i02 ';' J4, J2=i01 ';' J3;
A14: GCD-Algorithm=I13 ';' J14 by SCMPDS_4:13;
  then
A15: GCD-Algorithm=I12 ';' J13 by SCMPDS_4:12;
  then
A16: GCD-Algorithm=I11 ';' J12 by SCMPDS_4:12;
  then
A17: GCD-Algorithm=I10 ';' J11 by SCMPDS_4:12;
  then
A18: GCD-Algorithm=I9 ';' J10 by SCMPDS_4:12;
  then
A19: GCD-Algorithm=I8 ';' J9 by SCMPDS_4:12;
  then
A20: GCD-Algorithm=I7 ';' J8 by SCMPDS_4:12;
  then
A21: GCD-Algorithm=I6 ';' J7 by SCMPDS_4:12;
  then
A22: GCD-Algorithm=I5 ';' J6 by SCMPDS_4:12;
  then
A23: GCD-Algorithm=I4 ';' J5 by SCMPDS_4:12;
  then
A24: GCD-Algorithm=I3 ';' J4 by SCMPDS_4:12;
  then
A25: GCD-Algorithm=I2 ';' J3 by SCMPDS_4:12;
  then GCD-Algorithm=i00 ';' J2 by SCMPDS_4:16;
  hence GCD-Algorithm. 0=i00 by SCMPDS_6:7;
A26: card Load i00=1 by COMPOS_1:54;
  GCD-Algorithm=Load i00 ';' i01 ';' J3 by A25,SCMPDS_4:9;
  hence GCD-Algorithm. 1=i01 by A26,Th7;
  thus GCD-Algorithm. 2=i02 by A1,A24,Th7;
  thus GCD-Algorithm. 3=i03 by A2,A23,Th7;
  thus GCD-Algorithm. 4=i04 by A3,A22,Th7;
  thus GCD-Algorithm. 5=i05 by A4,A21,Th7;
  thus GCD-Algorithm. 6=i06 by A5,A20,Th7;
  thus GCD-Algorithm. 7=i07 by A6,A19,Th7;
  thus GCD-Algorithm. 8=i08 by A7,A18,Th7;
  thus GCD-Algorithm. 9=i09 by A8,A17,Th7;
  thus GCD-Algorithm. 10=i10 by A9,A16,Th7;
  thus GCD-Algorithm. 11=i11 by A10,A15,Th7;
  thus GCD-Algorithm. 12=i12 by A11,A14,Th7;
  GCD-Algorithm=I14 ';' Load i14 by SCMPDS_4:def 3;
  hence GCD-Algorithm. 13=i13 by A12,Th7;
  thus thesis by A13,Th6;
end;
