:: deftheorem defines Span JORDAN13:def 1 :
for C being non empty non horizontal non vertical being_simple_closed_curve Subset of (TOP-REAL 2)
for n being Nat st n is_sufficiently_large_for C holds
for b₃ being non constant standard clockwise_oriented special_circular_sequence holds
( b₃ = Span (C,n) iff ( b₃ is_sequence_on Gauge (C,n) & b₃ /. 1 = (Gauge (C,n)) * ((X-SpanStart (C,n)),(Y-SpanStart (C,n))) & b₃ /. 2 = (Gauge (C,n)) * (((X-SpanStart (C,n)) -' 1),(Y-SpanStart (C,n))) & ( for k being Nat st 1 <= k & k + 2 <= len b₃ holds
( ( front_right_cell (b₃,k,(Gauge (C,n))) misses C & front_left_cell (b₃,k,(Gauge (C,n))) misses C implies b₃ turns_left k, Gauge (C,n) ) & ( front_right_cell (b₃,k,(Gauge (C,n))) misses C & front_left_cell (b₃,k,(Gauge (C,n))) meets C implies b₃ goes_straight k, Gauge (C,n) ) & ( front_right_cell (b₃,k,(Gauge (C,n))) meets C implies b₃ turns_right k, Gauge (C,n) ) ) ) ) );