now

let x, y, z be Element of ; :: thesis: ( ( dist x,y = 0 implies x = y ) & ( x = y implies dist x,y = 0 ) & dist x,y = dist y,x & dist x,z <= (dist x,y) + (dist y,z) )
reconsider x' = x, y' = y, z' = z as Element of [:REAL ,REAL :] ;
A1: dist x,y = taxi_dist2 . x',y' by METRIC_1:def 1;
hence ( dist x,y = 0 iff x = y ) by Th19; :: thesis: ( dist x,y = dist y,x & dist x,z <= (dist x,y) + (dist y,z) )
dist y,x = taxi_dist2 . y',x' by METRIC_1:def 1;
hence dist x,y = dist y,x by A1, Th20; :: thesis: dist x,z <= (dist x,y) + (dist y,z)
( dist x,z = taxi_dist2 . x',z' & dist y,z = taxi_dist2 . y',z' ) by METRIC_1:def 1;
hence dist x,z <= (dist x,y) + (dist y,z) by A1, Th21; :: thesis: verum

end;

hence MetrStruct(# [:REAL ,REAL :],taxi_dist2 #) is non empty strict MetrSpace by METRIC_1:6; :: thesis: verum