environ vocabulary PARTFUN1, SEQ_1, FCONT_1, RELAT_1, LIMFUNC3, SEQ_2, ORDINAL2, BOOLE, FUNCT_1, LIMFUNC2, LIMFUNC1, RCOMP_1, ARYTM_1, FDIFF_1, ARYTM_3, SEQM_3, ARYTM; notation TARSKI, XBOOLE_0, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, NAT_1, SEQ_1, SEQ_2, SEQM_3, RELSET_1, PARTFUN1, RFUNCT_1, RCOMP_1, RFUNCT_2, FCONT_1, FDIFF_1, LIMFUNC1, LIMFUNC2, LIMFUNC3; constructors REAL_1, NAT_1, SEQ_2, SEQM_3, PROB_1, RFUNCT_1, RCOMP_1, RFUNCT_2, FCONT_1, FDIFF_1, LIMFUNC1, LIMFUNC2, LIMFUNC3, PARTFUN1, MEMBERED, XBOOLE_0; clusters FCONT_3, RELSET_1, XREAL_0, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0; requirements NUMERALS, SUBSET, BOOLE, ARITHM; begin reserve f,g for PartFunc of REAL,REAL, r,r1,r2,g1,g2,g3,g4,g5,g6,x,x0,t,c for Real, a,b,s for Real_Sequence, n,k for Nat; theorem :: L_HOSPIT:1 f is_continuous_in x0 & (for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f & g2<r2 & x0<g2 & g2 in dom f) implies f is_convergent_in x0; theorem :: L_HOSPIT:2 f is_right_convergent_in x0 & lim_right(f,x0) = t iff (for r st x0<r ex t st t<r & x0<t & t in dom f) & for a st a is convergent & lim a = x0 & rng a c= dom f /\ right_open_halfline(x0) holds f*a is convergent & lim(f*a) = t; theorem :: L_HOSPIT:3 f is_left_convergent_in x0 & lim_left(f,x0) = t iff (for r st r<x0 ex t st r<t & t<x0 & t in dom f) & for a st a is convergent & lim a = x0 & rng a c= dom f /\ left_open_halfline(x0) holds f*a is convergent & lim(f*a) = t; theorem :: L_HOSPIT:4 (ex N being Neighbourhood of x0 st N\{x0} c=dom f) implies (for r1,r2 st r1<x0 & x0<r2 ex g1,g2 st r1<g1 & g1<x0 & g1 in dom f & g2<r2 & x0<g2 & g2 in dom f); theorem :: L_HOSPIT:5 (ex N being Neighbourhood of x0 st f is_differentiable_on N & g is_differentiable_on N & N \ {x0} c= dom (f/g) & N c= dom ((f`|N)/(g`|N)) & f.x0 = 0 & g.x0 = 0 & (f`|N)/(g`|N) is_divergent_to+infty_in x0) implies f/g is_divergent_to+infty_in x0; theorem :: L_HOSPIT:6 (ex N being Neighbourhood of x0 st f is_differentiable_on N & g is_differentiable_on N & N \ {x0} c= dom (f/g) & N c= dom ((f`|N)/(g`|N)) & f.x0 = 0 & g.x0 = 0 & (f`|N)/(g`|N) is_divergent_to-infty_in x0) implies f/g is_divergent_to-infty_in x0; theorem :: L_HOSPIT:7 (ex r st r>0 & f is_differentiable_on ].x0,x0+r.[ & g is_differentiable_on ].x0,x0+r.[ & ].x0,x0+r.[ c= dom (f/g) & [.x0,x0+r.] c= dom ((f`|(].x0,x0+r.[))/(g`|(].x0,x0+r.[))) & f.x0 = 0 & g.x0 = 0 & f is_continuous_in x0 & g is_continuous_in x0 & (f`|(].x0,x0+r.[))/(g`|(].x0,x0+r.[)) is_right_convergent_in x0) implies f/g is_right_convergent_in x0 & ex r st r>0 & lim_right(f/g,x0) = lim_right(((f`|(].x0,x0+r.[))/(g`|(].x0,x0+r.[))),x0); theorem :: L_HOSPIT:8 (ex r st r>0 & f is_differentiable_on ].x0-r,x0.[ & g is_differentiable_on ].x0-r,x0.[ & ].x0-r,x0.[ c= dom (f/g) & [.x0-r,x0.] c= dom ((f`|(].x0-r,x0.[))/(g`|(].x0-r,x0.[))) & f.x0 = 0 & g.x0 = 0 & f is_continuous_in x0 & g is_continuous_in x0 & (f`|(].x0-r,x0.[))/(g`|(].x0-r,x0.[)) is_left_convergent_in x0) implies f/g is_left_convergent_in x0 & ex r st r>0 & lim_left(f/g,x0) = lim_left(((f`|(].x0-r,x0.[))/(g`|(].x0-r,x0.[))),x0); theorem :: L_HOSPIT:9 (ex N being Neighbourhood of x0 st f is_differentiable_on N & g is_differentiable_on N & N \ {x0} c= dom (f/g) & N c= dom ((f`|N)/(g`|N)) & f.x0 = 0 & g.x0 = 0 & (f`|N)/(g`|N) is_convergent_in x0) implies f/g is_convergent_in x0 & (ex N being Neighbourhood of x0 st lim(f/g,x0) = lim(((f`|N)/(g`|N)),x0)); theorem :: L_HOSPIT:10 (ex N being Neighbourhood of x0 st f is_differentiable_on N & g is_differentiable_on N & N \ {x0} c= dom (f/g) & N c= dom ((f`|N)/(g`|N)) & f.x0 = 0 & g.x0 = 0 & (f`|N)/(g`|N) is_continuous_in x0) implies f/g is_convergent_in x0 & lim(f/g,x0) = diff(f,x0)/diff(g,x0);