reconsider s = 1 as Real ;
let x0, 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; :: thesis:
assume A4: x1 in dom f ; :: thesis:
assume |.(x1 - x0).| < s ; :: thesis:
f /. x1 = f . x1 by A4, PARTFUN1:def 6
.= f . x0 by A1, A2, A4, FUNCT_1:def 10
.= f /. x0 by A2, PARTFUN1:def 6 ;
hence |.((f /. x1) - (f /. x0)).| < r by A3; :: thesis:

end;