let f1, f2 be Function of D,REAL; :: thesis: ( ( for X being Element of D st X in D holds

f1 . X = Sum (f * (canFS (eqSupport (f,X)))) ) & ( for X being Element of D st X in D holds

f2 . X = Sum (f * (canFS (eqSupport (f,X)))) ) implies f1 = f2 )

assume that

A6: for X being Element of D st X in D holds

f1 . X = Sum (f * (canFS (eqSupport (f,X)))) and

A7: for X being Element of D st X in D holds

f2 . X = Sum (f * (canFS (eqSupport (f,X)))) ; :: thesis: f1 = f2

for X being object st X in D holds

f1 . X = f2 . X

f1 . X = Sum (f * (canFS (eqSupport (f,X)))) ) & ( for X being Element of D st X in D holds

f2 . X = Sum (f * (canFS (eqSupport (f,X)))) ) implies f1 = f2 )

assume that

A6: for X being Element of D st X in D holds

f1 . X = Sum (f * (canFS (eqSupport (f,X)))) and

A7: for X being Element of D st X in D holds

f2 . X = Sum (f * (canFS (eqSupport (f,X)))) ; :: thesis: f1 = f2

for X being object st X in D holds

f1 . X = f2 . X

proof

hence
f1 = f2
by FUNCT_2:12; :: thesis: verum
let X be object ; :: thesis: ( X in D implies f1 . X = f2 . X )

assume A8: X in D ; :: thesis: f1 . X = f2 . X

then reconsider Y = X as Element of D ;

thus f1 . X = Sum (f * (canFS (eqSupport (f,Y)))) by A8, A6

.= f2 . X by A8, A7 ; :: thesis: verum

end;assume A8: X in D ; :: thesis: f1 . X = f2 . X

then reconsider Y = X as Element of D ;

thus f1 . X = Sum (f * (canFS (eqSupport (f,Y)))) by A8, A6

.= f2 . X by A8, A7 ; :: thesis: verum