set XX = [:X,(the carrier of M \ {a}):] \/ {[X,a]};
let w1, w2 be Function of [:([:X,(the carrier of M \ {a}):] \/ {[X,a]}),([:X,(the carrier of M \ {a}):] \/ {[X,a]}):],REAL ; :: thesis: ( ( for x, y being Element of [:X,(the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w1 . x,y = dist x2,y2 ) & ( x1 <> y1 implies w1 . x,y = (dist x2,a) + (dist a,y2) ) ) ) & ( for x, y being Element of [:X,(the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w2 . x,y = dist x2,y2 ) & ( x1 <> y1 implies w2 . x,y = (dist x2,a) + (dist a,y2) ) ) ) implies w1 = w2 )
assume that
A12: for x, y being Element of [:X,(the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w1 . x,y = dist x2,y2 ) & ( x1 <> y1 implies w1 . x,y = (dist x2,a) + (dist a,y2) ) ) and
A13: for x, y being Element of [:X,(the carrier of M \ {a}):] \/ {[X,a]}
for x1, y1 being set
for x2, y2 being Point of M st x = [x1,x2] & y = [y1,y2] holds
( ( x1 = y1 implies w2 . x,y = dist x2,y2 ) & ( x1 <> y1 implies w2 . x,y = (dist x2,a) + (dist a,y2) ) ) ; :: thesis: w1 = w2

now

let x, y be set ; :: thesis: ( x in [:X,(the carrier of M \ {a}):] \/ {[X,a]} & y in [:X,(the carrier of M \ {a}):] \/ {[X,a]} implies w1 . x,y = w2 . x,y )
assume A14: ( x in [:X,(the carrier of M \ {a}):] \/ {[X,a]} & y in [:X,(the carrier of M \ {a}):] \/ {[X,a]} ) ; :: thesis: w1 . x,y = w2 . x,y
consider x1 being set , x2 being Point of M such that
A15: ( x = [x1,x2] & ( ( x1 in X & x2 <> a ) or ( x1 = X & x2 = a ) ) ) by A14, Th38;
consider y1 being set , y2 being Point of M such that
A16: ( y = [y1,y2] & ( ( y1 in X & y2 <> a ) or ( y1 = X & y2 = a ) ) ) by A14, Th38;
reconsider x2 = x2, y2 = y2 as Point of M ;
( x1 = y1 or x1 <> y1 ) ;
then ( ( w1 . x,y = dist x2,y2 & w2 . x,y = dist x2,y2 ) or ( w1 . x,y = (dist x2,a) + (dist a,y2) & w2 . x,y = (dist x2,a) + (dist a,y2) ) ) by A12, A13, A14, A15, A16;
hence w1 . x,y = w2 . x,y ; :: thesis: verum

end;

hence w1 = w2 by BINOP_1:1; :: thesis: verum