let seq1 be without-infty ExtREAL_sequence; :: thesis: for seq2 being without+infty ExtREAL_sequence holds
( ( seq1 is convergent_to_+infty & seq2 is convergent_to_-infty implies ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty ) ) & ( seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty ) ) & ( seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_-infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_+infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = -infty & lim (seq2 - seq1) = +infty ) ) & ( seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_finite_number & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_finite_number & seq2 - seq1 is convergent & lim (seq1 - seq2) = (lim seq1) - (lim seq2) & lim (seq2 - seq1) = (lim seq2) - (lim seq1) ) ) )
let seq2 be without+infty ExtREAL_sequence; :: thesis: ( ( seq1 is convergent_to_+infty & seq2 is convergent_to_-infty implies ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty ) ) & ( seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty ) ) & ( seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_-infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_+infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = -infty & lim (seq2 - seq1) = +infty ) ) & ( seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_finite_number & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_finite_number & seq2 - seq1 is convergent & lim (seq1 - seq2) = (lim seq1) - (lim seq2) & lim (seq2 - seq1) = (lim seq2) - (lim seq1) ) ) )
reconsider f1 = - seq1 as without+infty ExtREAL_sequence ;
reconsider f2 = - seq2 as without-infty ExtREAL_sequence ;

hereby :: thesis: ( ( seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty ) ) & ( seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_-infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_+infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = -infty & lim (seq2 - seq1) = +infty ) ) & ( seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_finite_number & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_finite_number & seq2 - seq1 is convergent & lim (seq1 - seq2) = (lim seq1) - (lim seq2) & lim (seq2 - seq1) = (lim seq2) - (lim seq1) ) ) )

assume A1: ( seq1 is convergent_to_+infty & seq2 is convergent_to_-infty ) ; :: thesis: ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty )
then A2: ( f1 is convergent_to_-infty & f2 is convergent_to_+infty ) by Th17;
then reconsider F1 = f1 + seq2, F2 = seq1 + f2 as convergent ExtREAL_sequence by A1, Th24, Th18;
A3: ( f1 + seq2 is convergent_to_-infty & f1 + seq2 is convergent & lim (f1 + seq2) = -infty & seq1 + f2 is convergent_to_+infty & seq1 + f2 is convergent & lim (seq1 + f2) = +infty ) by A1, A2, Th24, Th18;
A4: seq1 - seq2 = seq1 + (- seq2) by Th10
.= - (seq2 - seq1) by Th10
.= - ((- seq1) + seq2) by Th10 ;
seq2 - seq1 = seq2 + (- seq1) by Th10
.= - (seq1 - seq2) by Th10
.= - ((- seq2) + seq1) by Th10 ;
hence ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty ) by A3, A4, Th17, XXREAL_3:5, XXREAL_3:6; :: thesis: verum

end;

hereby :: thesis: ( ( seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_-infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_+infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = -infty & lim (seq2 - seq1) = +infty ) ) & ( seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_finite_number & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_finite_number & seq2 - seq1 is convergent & lim (seq1 - seq2) = (lim seq1) - (lim seq2) & lim (seq2 - seq1) = (lim seq2) - (lim seq1) ) ) )

assume A1: ( seq1 is convergent_to_+infty & seq2 is convergent_to_finite_number ) ; :: thesis: ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty )
then A2: ( f1 is convergent_to_-infty & f2 is convergent_to_finite_number ) by Th17;
then reconsider F1 = f1 + seq2, F2 = seq1 + f2 as convergent ExtREAL_sequence by A1, Th24, Th19;
A3: ( f1 + seq2 is convergent_to_-infty & f1 + seq2 is convergent & lim (f1 + seq2) = -infty & seq1 + f2 is convergent_to_+infty & seq1 + f2 is convergent & lim (seq1 + f2) = +infty ) by A1, A2, Th24, Th19;
A4: seq1 - seq2 = seq1 + (- seq2) by Th10
.= - (seq2 - seq1) by Th10
.= - ((- seq1) + seq2) by Th10 ;
seq2 - seq1 = seq2 + (- seq1) by Th10
.= - (seq1 - seq2) by Th10
.= - ((- seq2) + seq1) by Th10 ;
hence ( seq1 - seq2 is convergent_to_+infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_-infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = +infty & lim (seq2 - seq1) = -infty ) by A3, A4, Th17, XXREAL_3:5, XXREAL_3:6; :: thesis: verum

end;

hereby :: thesis: ( seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number implies ( seq1 - seq2 is convergent_to_finite_number & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_finite_number & seq2 - seq1 is convergent & lim (seq1 - seq2) = (lim seq1) - (lim seq2) & lim (seq2 - seq1) = (lim seq2) - (lim seq1) ) )

assume A1: ( seq1 is convergent_to_-infty & seq2 is convergent_to_finite_number ) ; :: thesis: ( seq1 - seq2 is convergent_to_-infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_+infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = -infty & lim (seq2 - seq1) = +infty )
then A2: ( f1 is convergent_to_+infty & f2 is convergent_to_finite_number ) by Th17;
then reconsider F1 = f1 + seq2, F2 = seq1 + f2 as convergent ExtREAL_sequence by A1, Th24, Th22;
A3: ( f1 + seq2 is convergent_to_+infty & f1 + seq2 is convergent & lim (f1 + seq2) = +infty & seq1 + f2 is convergent_to_-infty & seq1 + f2 is convergent & lim (seq1 + f2) = -infty ) by A1, A2, Th24, Th22;
A4: seq1 - seq2 = seq1 + (- seq2) by Th10
.= - (seq2 - seq1) by Th10
.= - ((- seq1) + seq2) by Th10 ;
seq2 - seq1 = seq2 + (- seq1) by Th10
.= - (seq1 - seq2) by Th10
.= - ((- seq2) + seq1) by Th10 ;
hence ( seq1 - seq2 is convergent_to_-infty & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_+infty & seq2 - seq1 is convergent & lim (seq1 - seq2) = -infty & lim (seq2 - seq1) = +infty ) by A3, A4, Th17, XXREAL_3:5, XXREAL_3:6; :: thesis: verum

end;

assume A1: ( seq1 is convergent_to_finite_number & seq2 is convergent_to_finite_number ) ; :: thesis: ( seq1 - seq2 is convergent_to_finite_number & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_finite_number & seq2 - seq1 is convergent & lim (seq1 - seq2) = (lim seq1) - (lim seq2) & lim (seq2 - seq1) = (lim seq2) - (lim seq1) )
then A2: ( f1 is convergent_to_finite_number & f2 is convergent_to_finite_number ) by Th17;
then reconsider F1 = f1 + seq2, F2 = seq1 + f2 as convergent ExtREAL_sequence by A1, Th24, Th23;
A3: ( f1 + seq2 is convergent_to_finite_number & f1 + seq2 is convergent & seq1 + f2 is convergent_to_finite_number & seq1 + f2 is convergent & lim (f1 + seq2) = (lim f1) + (lim seq2) & lim (seq1 + f2) = (lim seq1) + (lim f2) ) by A1, A2, Th24, Th23;
A4: seq1 - seq2 = seq1 + (- seq2) by Th10
.= - (seq2 - seq1) by Th10
.= - ((- seq1) + seq2) by Th10 ;
then A5: lim (seq1 - seq2) = - ((lim f1) + (lim seq2)) by A3, Th17;
A6: seq2 - seq1 = seq2 + (- seq1) by Th10
.= - (seq1 - seq2) by Th10
.= - ((- seq2) + seq1) by Th10 ;
then A7: lim (seq2 - seq1) = - ((lim f2) + (lim seq1)) by A3, Th17;
( seq1 = - f1 & seq2 = - f2 ) by Th2;
then ( lim seq1 = - (lim f1) & lim seq2 = - (lim f2) ) by A2, Th17;
hence ( seq1 - seq2 is convergent_to_finite_number & seq1 - seq2 is convergent & seq2 - seq1 is convergent_to_finite_number & seq2 - seq1 is convergent & lim (seq1 - seq2) = (lim seq1) - (lim seq2) & lim (seq2 - seq1) = (lim seq2) - (lim seq1) ) by A3, A4, A5, A6, A7, Th17, XXREAL_3:9; :: thesis: verum