reconsider s = 1 as real number ;
let x0 be real number ; :: thesis:
let r be Real; :: thesis:
assume that
A2: x0 in dom f and
A3: 0 < r ; :: thesis:
take s = s; :: thesis:
thus 0 < s ; :: thesis:
let x1 be real number ; :: thesis:
assume A4: x1 in dom f ; :: thesis:
assume abs (x1 - x0) < s ; :: thesis:
f /. x1 = f . x1 by PARTFUN1:def 8, A4
.= f . x0 by A1, A2, A4, FUNCT_1:def 16
.= f /. x0 by PARTFUN1:def 8, A2 ;
hence ||.((f /. x1) - (f /. x0)).|| < r by A3, NORMSP_1:10; :: thesis:

end;